Search for Search... Log in Registration Chapter 4 Resource Masters Page: 1 of 79 Page Width Similar documents Answer Key for California State Standards: Algebra I Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences. More information Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation More information Vocabulary Words and De nitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coe cient Combine Like Terms More information SHARE HTML Algebra I Vocabulary Cards DOWNLOAD Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression Save this PDF as: WORD PNG TXT JPG 0 More information Ira Sharp 3 years ago Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Views: 199 Transcription 1 Chapter Resource Masters New York, New York Columbus, Ohio Woodland Hills, California Peoria, Illinois 2 StudentWorks TM This CD-ROM includes the entire Student Edition along with the Study Guide, Practice, and Enrichment masters. TeacherWorks TM All of the materials found in this booklet are included for viewing and printing in the Advanced Mathematical Concepts TeacherWorks CD-ROM. Copyright The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe Advanced Mathematical Concepts. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 0-07 ISBN: Advanced Mathematical Concepts Chapter Resource Masters XXX 3 Vocabulary Builder vii-ix Lesson - Study Guide Practice Enrichment Lesson - Study Guide Practice Enrichment Lesson - Study Guide Practice Enrichment Lesson - Study Guide Practice Enrichment Lesson -5 Study Guide Practice Enrichment Contents Lesson -7 Study Guide Practice Enrichment Lesson -8 Study Guide Practice Enrichment Chapter Assessment Chapter Test, Form A Chapter Test, Form B Chapter Test, Form C Chapter Test, Form A Chapter Test, Form B Chapter Test, Form C Chapter Extended Response Assessment Chapter Mid-Chapter Test Chapter Quizzes A & B Chapter Quizzes C & D Chapter SAT and ACT Practice Chapter Cumulative Review Unit Review Unit Test Lesson -6 Study Guide Practice Enrichment SAT and ACT Practice Answer Sheet, 0 Questions A SAT and ACT Practice Answer Sheet, 0 Questions A ANSWERS A-A0 Glencoe/McGraw-Hill iii Advanced Mathematical Concepts 4 A Teacher s Guide to Using the Chapter Resource Masters The Fast File Chapter Resource system allows you to conveniently le the resources you use most often. The Chapter Resource Masters include the core materials needed for Chapter. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet. All of the materials found in this booklet are included for viewing and printing in the Advanced Mathematical Concepts TeacherWorks CD-ROM. Vocabulary Builder Pages vii-x include a student study tool that presents the key vocabulary terms from the chapter. Students are to record de nitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. When to Use Give these pages to students before beginning Lesson -. Remind them to add de nitions and examples as they complete each lesson. Practice There is one master for each lesson. These problems more closely follow the structure of the Practice section of the Student Edition exercises. These exercises are of average di culty. When to Use These provide additional practice options or may be used as homework for second day teaching of the lesson. Study Guide There is one Study Guide master for each lesson. When to Use Use these masters as reteaching activities for students who need additional reinforcement. These pages can also be used in conjunction with the Student Edition as an instructional tool for those students who have been absent. Enrichment There is one master for each lesson. These activities may extend the concepts in the lesson, o er a historical or multicultural look at the concepts, or widen students perspectives on the mathematics they are learning. These are not written exclusively for honors students, but are accessible for use with all levels of students. When to Use These may be used as extra credit, short-term projects, or as activities for days when class periods are shortened. Glencoe/McGraw-Hill iv Advanced Mathematical Concepts 5 Assessment Options The assessment section of the Chapter Resources Masters o ers a wide range of assessment tools for intermediate and nal assessment. The following lists describe each assessment master and its intended use. Intermediate Assessment A Mid-Chapter Test provides an option to assess the rst half of the chapter. It is composed of free-response questions. Four free-response quizzes are included to o er assessment at appropriate intervals in the chapter. Chapter Assessments Chapter Tests Forms A, B, and C Form tests contain multiple-choice questions. Form A is intended for use with honors-level students, Form B is intended for use with averagelevel students, and Form C is intended for use with basic-level students. These tests are similar in format to o er comparable testing situations. Forms A, B, and C Form tests are composed of free-response questions. Form A is intended for use with honors-level students, Form B is intended for use with average-level students, and Form C is intended for use with basic-level students. These tests are similar in format to o er comparable testing situations. All of the above tests include a challenging Bonus question. The Extended Response Assessment includes performance assessment tasks that are suitable for all students. A scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. Continuing Assessment The SAT and ACT Practice o ers continuing review of concepts in various formats, which may appear on standardized tests that they may encounter. This practice includes multiple-choice, quantitativecomparison, and grid-in questions. Bubblein and grid-in answer sections are provided on the master. The Cumulative Review provides students an opportunity to reinforce and retain skills as they proceed through their study of advanced mathematics. It can also be used as a test. The master includes freeresponse questions. Answers Page A is an answer sheet for the SAT and ACT Practice questions that appear in the Student Edition on page 7. Page A is an answer sheet for the SAT and ACT Practice master. These improve students familiarity with the answer formats they may encounter in test taking. The answers for the lesson-by-lesson masters are provided as reduced pages with answers appearing in red. Full-size answer keys are provided for the assessment options in this booklet. Glencoe/McGraw-Hill v Advanced Mathematical Concepts 6 Chapter Leveled Worksheets Glencoe s leveled worksheets are helpful for meeting the needs of every student in a variety of ways. These worksheets, many of which are found in the FAST FILE Chapter Resource Masters, are shown in the chart below. Study Guide masters provide worked-out examples as well as practice problems. Each chapter s Vocabulary Builder master provides students the opportunity to write out key concepts and de nitions in their own words. Practice masters provide average-level problems for students who are moving at a regular pace. Enrichment masters o er students the opportunity to extend their learning. Five Di erent Options to Meet the Needs of Every Student in a Variety of Ways primarily skills primarily concepts primarily applications BASIC AVERAGE ADVANCED Study Guide Vocabulary Builder Parent and Student Study Guide (online) Practice 5 Enrichment Glencoe/McGraw-Hill vi Advanced Mathematical Concepts 7 Chapter Reading to Learn Mathematics Vocabulary Builder This is an alphabetical list of the key vocabulary terms you will learn in Chapter. As you study the chapter, complete each term s de nition or description. Remember to add the page number where you found the term. Vocabulary Term completing the square Found on Page De nition/Description/Example complex number conjugate degree depressed polynomial Descartes Rule of Signs discriminant extraneous solution Factor Theorem Fundamental Theorem of Algebra (continued on the next page) Glencoe/McGraw-Hill vii Advanced Mathematical Concepts 8 Chapter Reading to Learn Mathematics Vocabulary Builder (continued) Vocabulary Term imaginary number Found on Page De nition/Description/Example Integral Root Theorem leading coe cient Location Principle lower bound lower Bound Theorem partial fractions polynomial equation polynomial function polynomial in one variable pure imaginary number (continued on the next page) Glencoe/McGraw-Hill viii Advanced Mathematical Concepts 9 Chapter Reading to Learn Mathematics Vocabulary Builder (continued) Vocabulary Term Quadratic Formula Found on Page De nition/Description/Example radical equation radical inequality rational equation rational inequality Rational Root theorem Remainder Theorem synthetic division upper bound Upper Bound Theorem zero Glencoe/McGrawHill ix Advanced Mathematical Concepts 10 BLANK 11 - Study Guide Polynomial Functions The degree of a polynomial in one variable is the greatest exponent of its variable. The coe cient of the variable with the greatest exponent is called the leading coe cient. If a function ƒ(x) is de ned by a polynomial in one variable, then it is a polynomial function. The values of x for which ƒ(x) 0 are called the zeros of the function. Zeros of the function are roots of the polynomial equation when ƒ(x) 0. A polynomial equation of degree n has exactly n roots in the set of complex numbers. Example Example Example State the degree and leading coe cient of the polynomial function ƒ(x) 6x 5 8x 8x. Then determine whether is a zero of ƒ(x). 6x 5 8x 8x has a degree of 5 and a leading coe cient of 6. Evaluate the function for x ƒ That is, nd ƒ. x Since ƒ 0, is a zero of ƒ(x) 6x5 8x 8x. Write a polynomial equation of least degree with roots 0, i, and i. The linear factors for the polynomial are x 0, x i, and x i. Find the products of these factors. (x 0)(x i)(x i) 0 x(x i ) 0 x(x ) 0 i () or x x 0 State the number of complex roots of the equation x x 0. Then nd the roots. The polynomial has a degree of, so there are two complex roots. Factor the equation to nd the roots. x x 0 (x )(x ) 0 To nd each root, set each factor equal to zero. x 0 x 0 x x x The roots are and. Glencoe/McGraw-Hill Advanced Mathematical Concepts 12 - Practice Polynomial Functions State the degree and leading coe cient of each polynomial.. 6a a a. p 7p 5 p 5 Write a polynomial equation of least degree for each set of roots.., 0.5,.,,,, 5. i,, 6., i, i State the number of complex roots of each equation. Then nd the roots and graph the related function. 7. x x 0 9. c c 0 0. x x 5x 0. Real Estate A developer wants to build homes on a rectangular plot of land kilometers long and kilometers wide. In this part of the city, regulations require a greenbelt of uniform width along two adjacent sides. The greenbelt must be 0 times the area of the development. Find the width of the greenbelt. Glencoe/McGraw-Hill Advanced Mathematical Concepts 13 - and h(x) x 8. Enrichment Graphic Addition One way to sketch the graphs of some polynomial functions is to use addition of ordinates. This method is useful when a polynomial function f (x) can be written as the sum of two other functions, g(x) and h(x), that are easier to graph. Then, each f (x) can be found by mentally adding the corresponding g(x) and h(x). The graph at the right shows how to construct the graph of f(x) x x 8 from the graphs of g(x) x In each problem, the graphs of g(x) and h(x) are shown. Use addition of ordinates to graph a new polynomial function f(x), such that f(x) g(x) h(x). Then write the equation for f(x) Glencoe/McGraw-Hill Advanced Mathematical Concepts 14 - Study Guide Quadratic Equations A quadratic equation is a polynomial equation with a degree of. Solving quadratic equations by graphing usually does not yield exact answers. Also, some quadratic expressions are not factorable. However, solutions can always be obtained by completing the square. Example Solve x x 7 0 by completing the square. x x 7 0 x x 7 Subtract 7 from each side. x x Complete the square by adding (), or 6, to each side. (x 6) 9 Factor the perfect square trinomial. x 6 9 Take the square root of each side. x 6 9 Add 6 to each side. The roots of the equation are 6 9. Completing the square can be used to develop a general formula for solving any quadratic equation of the form ax bx c 0. This formula is called the Quadratic Formula and can be used to nd the roots of any quadratic equation. Quadratic Formula If ax b b bx c 0 with a 0, x ac. a In the Quadratic Formula, the radicand b ac is called the discriminant of the equation. The discriminant tells the nature of the roots of a quadratic equation or the zeros of the related quadratic function. Example Find the discriminant of x x 7 and describe the nature of the roots of the equation. Then solve the equation by using the Quadratic Formula. Rewrite the equation using the standard form ax bx c 0. x x 7 0 a, b, and c 7 The value of the discriminant b ac is () ()(7), or 65. Since the value of the discriminant is greater than zero, there are two distinct real roots. Now substitute the coe cients into the quadratic formula and solve. b b x x ac () () ()( 7) () a x 65 The roots are 65 and 65. Glencoe/McGraw-Hill Advanced Mathematical Concepts 15 - Practice Quadratic Equations Solve each equation by completing the square.. x 5x 0. x x 7 Find the discriminant of each equation and describe the nature of the roots of the equation. Then solve the equation by using the Quadratic Formula.. x x 6 0. x x x x 0 6. x x 5 0 Solve each equation. 7. x 5x x x 0 9. Architecture The ancient Greek mathematicians thought that the most pleasing geometric forms, such as the ratio of the height to the width of a doorway, were created using the golden section. However, they were surprised to learn that the golden section is not a rational number. One way of expressing the golden section is by using a line segment. In the line segment shown, A B AC. AC C If B AC unit, nd the ratio A B AC. Glencoe/McGraw-Hill 5 Advanced Mathematical Concepts 16 - Enrichment Conjugates and Absolute Value When studying complex numbers, it is often convenient to represent a complex number by a single variable. For example, we might let z x yi. We denote the conjugate of z by z. Thus, z x yi. We can de ne the absolute value of a complex number as follows. z x yi x y There are many important relationships involving conjugates and absolute values of complex numbers. Example Show that z zz for any complex number z. Let z x yi. Then, zz (x yi)(x yi) x y x y z z Example Show that is the multiplicative inverse for any z nonzero complex number z. We know that z zz. If z 0, then we have z z z. Thus, is the multiplicative z z inverse of z. For each of the following complex numbers, nd the absolute value and multiplicative inverse.. i. i. 5i. 5 i 5. i 6. i 7. i 8. i 9. i Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts 17 - Study Guide The Remainder and Factor Theorems The Remainder Theorem If a polynomial P(x) is divided by x r, the remainder is a constant P(r ), and P(x) (x r ) Q(x) P(r ) where Q(x) is a polynomial with degree one less than the degree of P(x). Example Divide x 5x 7x by x. x x x 9 x x 0x 5x 7x x x x 5x x 9x x 7x x x 9x 9x remainder Find the value of r in this division. x r x r r According to the Remainder Theorem, P(r) or P() should equal 75. Use the Remainder Theorem to check the remainder found by long division. P(x) x 5x 7x P() () 5() 7() or 75 The Factor Theorem is a special case of the Remainder Theorem and can be used to quickly test for factors of a polynomial. The Factor Theorem Example The binomial x r is a factor of the polynomial P(x) if and only if P(r) 0. Use the Remainder Theorem to nd the remainder when x 5x x 8 is divided by x. State whether the binomial is a factor of the polynomial. Explain. Find ƒ() to see if x is a factor. ƒ(x) x 5x x 8 ƒ() () 5() () Since ƒ() 0, the remainder is 0. So the binomial x is a factor of the polynomial by the Factor Theorem. Glencoe/McGraw-Hill 7 Advanced Mathematical Concepts 18 - Practice The Remainder and Factor Theorems Divide using synthetic division.. (x x ) (x 5). (x 5x ) (x ). (x x ) (x ). (x x 8x ) (x ) Use the Remainder Theorem to nd the remainder for each division. State whether the binomial is a factor of the polynomial. 5. (x x x 9) (x ) 6. (x x 0x ) (x ) 7. (t 0t t 5) (t ) 8. (0x x 7x 0) (x ) 9. (x 5x x ) (x ) 0. (x x x x) (x 7). ( y y 0) ( y ). (n n 0n n ) (n ). Use synthetic division to nd all the factors of x 6x 9x 5 if one of the factors is x.. Manufacturing A cylindrical chemical storage tank must have a height meters greater than the radius of the top of the tank. Determine the radius of the top and the height of the tank if the tank must have a volume of 5.7 cubic meters. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts 19 - Enrichment The Secret Cubic Equation You might have supposed that there existed simple formulas for solving higher-degree equations. After all, there is a simple formula for solving quadratic equations. Might there not be formulas for cubics, quartics, and so forth? There are formulas for some higher-degree equations, but they are certainly not simple formulas! Here is a method for solving a reduced cubic of the form x ax b 0 published by Jerome Cardan in 55. Cardan was given the formula by another mathematician, Tartaglia. Tartaglia made Cardan promise to keep the formula secret, but Cardan published it anyway. He did, however, give Tartaglia the credit for inventing the formula! Let R b Then, x b R b R a 7 Use Cardan s method to nd the real root of each cubic equation. Round answers to three decimal places. Then sketch a graph of the corresponding function on the grid provided.. x 8x 0. x x 5 0. x x 0. x x 0 Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts 20 - Study Guide The Rational Root Theorem The Rational Root Theorem provides a means of determining possible rational roots of an equation. Descartes Rule of Signs can be used to determine the possible number of positive real zeros and the possible number of negative real zeros. Rational Root Theorem Let a 0 x n a x n... a n x a n 0 represent a polynomial equation of degree n with integral coe cients. If a rational number p, where p and q have no common factors, is a root of the equation, then q p is a factor of a n and q is a factor of a 0. Example List the possible rational roots of x 5x 7x 6 0. Then determine the rational roots. p is a factor of 6 and q is a factor of possible values of p:,,, 6 possible values of q: possible rational roots, p :,,, 6 q Test the possible roots using synthetic division. r There is a root at x. The depressed polynomial is x 7x. You can use the Quadratic Formula to nd the two irrational roots. Example Find the number of possible positive real zeros and the number of possible negative real zeros for f(x) x x x 8x 8. According to Descartes Rule of Signs, the number of positive real zeros is the same as the number of sign changes of the coe cients of the terms in descending order or is less than this by an even number. Count the sign changes. ƒ(x) x x x 8x There are three changes. So, there are or positive real zeros. The number of negative real zeros is the same as the number of sign changes of the coe cients of the terms of ƒ(x), or less than this number by an even number. ƒ(x) (x) (x) (x) 8(x) There is one change. So, there is negative real zero Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts 21 - Practice The Rational Root Theorem List the possible rational roots of each equation. Then determine the rational roots.. x x 8x 0. x x x 0. 6x x 0. x x 6x x x x x x x 0 7. x x 8x x x x 5 0 Find the number of possible positive real zeros and the number of possible negative real zeros. Then determine the rational zeros. 9. ƒ(x) x x 9x 0 0. ƒ(x) x x 7x x 6. Driving An automobile moving at meters per second on level ground begins to decelerate at a rate of.6 meters per second squared. The formula for the distance an object has traveled is d(t) v 0 t at, where v 0 is the initial velocity and a is the acceleration. For what value(s) of t does d(t) 0 meters? Glencoe/McGraw-Hill Advanced Mathematical Concepts 22 - Enrichment Scrambled Proofs The proofs on this page have been scrambled. Number the statements in each proof so that they are in a logical order. The Remainder Theorem Thus, if a polynomial f (x) is divided by x a, the remainder is f (a). In any problem of division the following relation holds: dividend quotient divisor remainder. In symbols,this may be written as: Equation () tells us that the remainder R is equal to the value f (a); that is, f (x) with a substituted for x. For x a, Equation () becomes: Equation () f (a) R, since the rst term on the right in Equation () becomes zero. Equation () f (x) Q(x)(x a) R, in which f (x) denotes the original polynomial, Q(x) is the quotient, and R the constant remainder. Equation () is true for all values of x, and in particular, it is true if we set x a. The Rational Root Theorem Each term on the left side of Equation () contains the factor a; hence, a must be a factor of the term on the right, namely, c n b n. But by hypothesis, a is not a factor of b unless a ±. Hence, a is a factor of c n. a f n c 0 a n c a n... c n a c 0 n Thus, in the polynomial equation given in Equation (), a is a factor of c n and b is a factor of c 0. In the same way, we can show that b is a factor of c 0. A polynomial equation with integral coe cients of the form Equation () f (x) c 0 x n c x n... c n x c n 0 a a has a rational root, where the fraction is reduced to lowest terms. Since a b b b is a root of f (x) 0, then b b If each side of this equation is multiplied by b n and the last term is transposed, it becomes b b Equation () c 0 a n c a n b... c n ab n c n b n Glencoe/McGraw-Hill Advanced Mathematical Concepts 23 -5 Study Guide Locating Zeros of a Polynomial Function A polynomial function may have real zeros that are not rational numbers. The Location Principle provides a means of locating and approximating real zeros. For the polynomial function y ƒ(x), if a and b are two numbers with ƒ(a) positive and ƒ(b) negative, then there must be at least one real zero between a and b. For example, if ƒ(x) x, ƒ(0) and ƒ(). Thus, a zero exists somewhere between 0 and. The Upper Bound Theorem and the Lower Bound Theorem are also useful in locating the zeros of a function and in determining whether all the zeros have been found. If a polynomial function P(x) is divided by x c, and the quotient and the remainder have no change in sign, c is an upper bound of the zeros of P(x). If c is an upper bound of the zeros of P(x), then c is a lower bound of the zeros of P(x). Example Example Determine between which consecutive integers the real zeros of ƒ(x) x x x 5 are located. According to Descartes Rule of Signs, there are two or zero positive real roots and one negative real root. Use synthetic division to evaluate ƒ(x) for consecutive integral values of x. r There is a zero at. The changes in sign indicate that there are also zeros between and and between and. This result is consistent with Descartes Rule of Signs. Use the Upper Bound Theorem to show that is an upper bound and the Lower Bound Theorem to show that is a lower bound of the zeros of ƒ(x) x x x. Synthetic division is the most e cient way to test potential upper and lower bounds. First, test for the upper bound. r 0 Since there is no change in the signs in the quotient and remainder, is an upper bound. Now, test for the lower bound of ƒ(x) by showing that is an upper bound of ƒ(x). ƒ(x) (x) (x) (x) x x x r 5 Since there is no change in the signs, is a lower bound of ƒ(x). Glencoe/McGraw-Hill Advanced Mathematical Concepts 24 -5 Practice Locating Zeros of a Polynomial Function Determine between which consecutive integers the real zeros of each function are located.. ƒ(x) x 0x x. ƒ(x) x 5x 7x. ƒ(x) x x x. ƒ(x) x x 7x 9 5. ƒ(x) x 6x 5x 96x 6 6. ƒ(x) x 9 Approximate the real zeros of each function to the nearest tenth. 7. ƒ(x) x x 8. ƒ(x) x x 9. ƒ(x) x 6x 0. ƒ(x) x x. ƒ(x) x x x. ƒ(x) x 5x Use the Upper Bound Theorem to nd an integral upper bound and the Lower Bound Theorem to nd an integral lower bound of the zeros of each function.. ƒ(x) x x 8x x 0. ƒ(x) x x x 6 5. For ƒ(x) x x, determine the number and type of possible complex zeros. Use the Location Principle to determine the zeros to the nearest tenth. The graph has a relative maximum at (0, 0) and a relative minimum at (, ). Sketch the graph. Glencoe/McGraw-Hill Advanced Mathematical Concepts 25 -5 Enrichment The Bisection Method for Approximating Real Zeros The bisection method can be used to approximate zeros of polynomial functions like f (x) x x x. Since f () and f (), there is at least one real zero between and. The midpoint of this interval is.5. Since f (.5).875, the zero is between.5 and. The midpoint of this interval is Since f (.75) 0.7, the zero is between.5 and and f (.65) 0.9. The zero is between.65 and.75. The midpoint of this interval is Since f (.6875) 0., the zero is between.6875 and.75. Therefore, the zero is.7 to the nearest tenth. The diagram below summarizes the bisection method. Using the bisection method, approximate to the nearest tenth the zero between the two integral values of each function.. f (x) x x x, f (0), f (). f (x) x x 5, f (), f (). f (x) x 5 x, f (), f (). f (x) x x 7, f ( ), f( ) 5 5. f (x) x x 7x 6, f (), f (5) 6 Glencoe/McGraw-Hill 5 Advanced Mathematical Concepts 26 -6 Study Guide Rational Equations and Partial Fractions A rational equation consists of one or more rational expressions. One way to solve a rational equation is to multiply each side of the equation by the least common denominator (LCD). Any possible solution that results in a zero in the denominator must be excluded from your list of solutions. In order to nd the LCD, it is sometimes necessary to factor the denominators. If a denominator can be factored, the expression can be rewritten as the sum of partial fractions. Example Solve x ( 5 x x ) 6 x. 6(x ) x ( x ) 6(x ) 5 x 6 x Multiply each side by the LCD, 6(x ). (x ) (x )(5x) 6() x 5x 0x 6 Simplify. 5x x 0 Write in standard form. (5x )(x ) 0 Factor. 5x 0 x 0 x x 5 Since x cannot equal because a zero denominator results, the only solution is 5. Example Decompose x x x into partial fractions. Factor the denominator and express the factored form as the sum of two fractions using A and B as numerators and the factors as denominators. x x (x )(x ) x A x x x B x x A(x ) B(x ) Let x. Let x. () A( ) () B( ) A 7 B A B 7 Example Solve t. t Rewrite the inequality as the related function ƒ(t) t. t Find the zeros of this function. t t t t t() t(0) 5 t 0 t.5 The zero is.5. The excluded value is 0. On a number line, mark these values with vertical dashed lines. Testing each interval shows the solution set to be 0 t.5. 7 x or x x (x 7 x x ) (x ) Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts 27 -6 Practice Rational Equations and Partial Fractions Solve each equation.. 5 m 8 0. m b b b b. 6n 9 n n n. t t 5. a a p a 6. 5 p p p p Decompose each expression into partial fractions. 7. x 9 8. x 7 x x x x Solve each inequality t t 0. n n n n y.. x 5x y. Commuting Rosea drives her car 0 kilometers to the train station, where she boards a train to complete her trip. The total trip is 0 kilometers. The average speed of the train is 0 kilometers per hour faster than that of the car. At what speed must she drive her car if the total time for the trip is less than.5 hours? Glencoe/McGraw-Hill 7 Advanced Mathematical Concepts 28 -6 Enrichment Inverses of Conditional Statements In the study of formal logic, the compound statement if p, then q where p and q represent any statements, is called a conditional or an implication. The symbolic representation of a conditional is p q. p q If the determinant of a matrix is 0, then the matrix does not have an inverse. If both p and q are negated, the resulting compound statement is called the inverse of the original conditional. The symbolic notation for the negation of p is p. Conditional Inverse If a conditional is true, its inverse p q p q may be either true or false. Example Find the inverse of each conditional. a. p q: If today is Monday, then tomorrow is Tuesday. (true) p q If today is not Monday, then tomorrow is not Tuesday. (true) b. p q If ABCD is a square, then ABCD is a rhombus. (true) p q If ABCD is not a square, then ABCD is not a rhombus. (false) Write the inverse of each conditional.. q p. p q. q p. If the base angles of a triangle are congruent, then the triangle is isosceles. 5. If the moon is full tonight, then we ll have frost by morning. Tell whether each conditional is true or false. Then write the inverse of the conditional and tell whether the inverse is true or false. 6. If this is October, then the next month is December. 7. If x > 5, then x > 6, x R. 8. If x 0, then x 0, x R. 9. Make a conjecture about the truth value of an inverse if the conditional is false. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts 29 -7 Study Guide Radical Equations and Inequalities Equations in which radical expressions include variables are known as radical equations. To solve radical equations, rst isolate the radical on one side of the equation. Then raise each side of the equation to the proper power to eliminate the radical expression. This process of raising each side of an equation to a power often introduces extraneous solutions. Therefore, it is important to check all possible solutions in the original equation to determine if any of them should be eliminated from the solution set. Radical inequalities are solved using the same techniques used for solving radical equations. Example Solve x x. x x x x Isolate the cube root. 6 x x Cube each side. 0 x x 6 0 (x 9)(x 7) Factor. x 9 0 x 7 0 x 9 x 7 Check both solutions to make sure they are not extraneous. x 9: x x x 7: x x (9) (9) (7) (7) 6 6 Example Solve x 5. x 5 (x 5) Square each side. x 5 Divide each side by. x x. In order for x 5 to be a real number, x 5 must be greater than or equal to zero. x 5 0 x 5 x.67 Since. is greater than.67, the solution is x.. Check this solution by testing values in the intervals de ned by the solution. Then graph the solution on a number line. Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts 30 -7 Practice Radical Equations and Inequalities Solve each equation.. x 6. x. 7r 5. 6x x 9 5. x x 6. 6n 7n 7. 5 x x x 8. r r Solve each inequality. 9. r 5 0. t 5. m 5. x 5 9. Engineering A team of engineers must design a fuel tank in the shape of a cone. The surface area of a cone (excluding the base) is given by the formula S r h. nd the radius of a cone with a height of meters and a surface area of 55 meters squared. Glencoe/McGraw-Hill 50 Advanced Mathematical Concepts 31 -7 Enrichment Discriminants and Tangents The diagram at the right shows that through a point P outside of a circle C, there are lines that do not intersect the circle, lines that intersect the circle in one point (tangents), and lines that intersect the circle in two points (secants). Given the coordinates for P and an equation for the circle C, how can we nd the equation of a line tangent to C that passes through P? Suppose P has coordinates P(0, 0) and C has equation (x ) y. Then a line tangent through P has equation y mx for some real number m. Thus, if T (r, s) is a point of tangency, then s mr and (r ) s. Therefore, (r ) (mr). r 8r 6 m r ( m )r 8r 6 ( m )r 8r 0 The equation above has exactly one real solution for r if the discriminant is 0, that is, when ( 8) ( m )() 0. Solve this equation for m and you will nd the slopes of the lines through P that are tangent to circle C.. a. Refer to the discussion above. Solve ( 8) ( m )() 0 to nd the slopes of the two lines tangent to circle C through point P. b. Use the values of m from part a to nd the coordinates of the two points of tangency.. Suppose P has coordinates (0, 0) and circle C has equation (x 9) y 9. Let m be the slope of the tangent line to C through P. a. Find the equations for the lines tangent to circle C through point P. b. Find the coordinates of the points of tangency. Glencoe/McGraw-Hill 5 Advanced Mathematical Concepts 32 -8 Study Guide Modeling Real-World Data with Polynomial Functions In order to model real-world data using polynomial functions, you must be able to identify the general shape of the graph of each type of polynomial function. Example Determine the type of polynomial function that could be used to represent the data in each scatter plot. a. y b. O x The scatter plot seems to change direction three times, so a quartic function would best t the scatter plot. The scatter plot seems to change direction two times, so a cubic function would best t the scatter plot. Example An oil tanker collides with another ship and starts leaking oil. The Coast Guard measures the rate of ow of oil from the tanker and obtains the data shown in the table. Use a graphing calculator to write a polynomial function to model the set of data. Clear the statistical memory and input the data. Adjust the window to an appropriate setting and graph the statistical data. The data appear to change direction one time, so a quadratic function will t the scatter plot. Press STAT, highlight CALC, and choose 5:QuadReg. Then enter nd [L], nd [L] ENTER. Rounding the coe cients to the nearest tenth, f(x) 0.x.8x 6. models the data. Since the value of the coe cient of determination r is very close to, the polynomial is an excellent t. Time Flow rate (hours) (00s of liters per hour) [0, 0] scl: by [0.5] scl: 5 Glencoe/McGraw-Hill 5 Advanced Mathematical Concepts 33 -8 Practice Modeling Real-World Data with Polynomial Functions Write a polynomial function to model each set of data.. The farther a planet is from the Sun, the longer it takes to complete an orbit. Distance (AU) Period (days) ,775 0,68 60,67 90,58 Source: Astronomy: Fundamentals and Frontiers, by Jastrow, Robert, and Malcolm H. Thompson.. The amount of food energy produced by farms increases as more energy is expended. The following table shows the amount of energy produced and the amount of energy expended to produce the food. Energy Input (Calories) Energy Output (Calories) Source: NSTA Energy-Environment Source Book.. The temperature of Earth s atmosphere varies with altitude. Altitude (km) Temperature (K) Source: Living in the Environment, by Miller G. Tyler.. Water quality varies with the season. This table shows the average hardness (amount of dissolved minerals) of water in the Missouri River measured at Kansas City, Missouri. Month Jan. Feb. Mar. April May June July Aug. Sept. Oct. Nov. Dec. Hardness (CaCO ppm) Source: The Encyclopedia of Environmental Science, 97. Glencoe/McGraw-Hill 5 Advanced Mathematical Concepts 34 -8 Enrichment Number of Paths For the gure and adjacency matrix shown at the right, the number of paths or circuits of length can be found by computing the following product. V V V V V 0 A V 0 V 0 0 V 0 0 A AA In row column, the entry in the product matrix means that there are paths of length between V and V. The paths are V V V and V V V. Similarly, in row column, the entry means there is only path of length between V and V. Name the paths of length between the following.. V and V. V and V. V and V For Exercises -6, refer to the gure below.. The number of paths of length is given by the product A A A or A. Find the matrix for paths of length. 5. How many paths of length are there between Atlanta and St. Louis? Name them. 6. How would you nd the number of paths of length between the cities? Glencoe/McGraw-Hill 5 Advanced Mathematical Concepts nd the remainder when. 6x 5 x 8x 6 is divided by x. State whether the binomial is a factor of the polynomial. A. 8; no B. 0; yes C. 700; yes D. 0; no. Solve t t 0 by completing the square.. A. 6 B. 0, C. 0, D.. Use synthetic division to divide 8x 0x x 8x by x.. A. 8x 8x x R B. 8x 8x x 6 R7 C. 8x 6x 8x 0 R9 D. 8x 8x x 8. Solve x 6 9x 0.. A., 6 B. 9 C., 6 D List the possible rational roots of x 7x x A.,,, 6, 7,,,,,, 7, B.,, 6, 7,,,, 7, C.,,, 6, 7,,,,, 7, D.,,, 6, 7,,,,, 7, 6. Determine the rational roots of 6x 5x x A.,, B.,, C.,, D.,, 7. Solve x 5 x x x x x. 8 x A. B., C. 9 D., Find the discriminant of x 9 x and describe the nature of the 8. roots of the equation. A. 56; exactly one real root B. 56; two distinct real roots C. 56; no real roots D. 56; two distinct real roots 9. Solve x 0 by using the Quadratic Formula. 9. A. i B. C. D. 0, 6 0. Determine between which consecutive integers one or more real 0. zeros of ƒ(x) x x x are located. A. no real zeros B. 0 and C. and D. and 0 Glencoe/McGraw-Hill 55 Advanced Mathematical Concepts Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used More information Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication More information Algebra 1 Course Title Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept More information Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...} More information Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions. Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables. More information MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) o ered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers, More information Algebra and Geometry Review (61 topics, no due date) Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties More information MATH 60 NOTEBOOK CERTIFICATIONS MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5 More information Algebra II End of Course Exam Answer Key Segment I. Scienti c Calculator Only Algebra II End of Course Exam Answer Key Segment I Scienti c Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials More information LAKE ELSINORE UNIFIED SCHOOL DISTRICT LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description: More information MATH 10034 Fundamental Mathematics IV MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/003 4/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials. More information ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola More information Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity: More information CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical More information SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). 498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1 More information POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a More information Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that nding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we More information SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic More information Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute uently and make reasonable estimates. 1. Simplify More information 35 Chapter Chapter Test, Form A Write the letter for the correct answer in the blank at the right of each problem.. Use the Remainder Theorem to 36 Chapter Chapter Test, Form A (continued). Find the value of k so that the remainder of (kx 6x ). (x ) is 0. A. 7 B. 9 C. 7 D. 9. Find the number of possible negative real zeros for. ƒ(x) 6 x x 5x x. A. or 0 B. or C. 0 D.. Solve 7x.. A. x B. x C. x D. x 7. Decompose 5x into partial fractions. x x A. x x B. x x C. x x D. x x 5. Solve 5 x x x x. x A. 0 x 5 B. 0 x 5 C. x 0, x 5 D. x 0, x Approximate the real zeros of ƒ(x) x x to the nearest tenth. 6. A. B.. C..5 D. no real zeros 7. Solve x x. 7. A. B. C. D. 8. Which polynomial function best models the set of data below? 8. x f(x) A. y 0.0x 0.5x 0.x 0.8 B. y 0.x 0.5x 0.x 0.8 C. y 0.x 0.5x 0.x 0.8 D. y 0.0x 0.5x 0.x Solve 6 x. A. x B. 5 x C. x 5, x D. x Find the polynomial equation of least degree with roots, i, 0. and i. A. x x x 6 0 B. x x x 6 0 C. x x x 6 0 D. x x x 6 0 Bonus Find the discriminant of ( )x ( )x. Bonus: A. B. 6 9 C. D. 7 Glencoe/McGraw-Hill 56 Advanced Mathematical Concepts 37 Chapter Chapter Test, Form B Write the letter for the correct answer in the blank at the right of each problem.. Use the Remainder Theorem to nd the remainder when. x 6x x is divided by x. State whether the binomial is a factor of the polynomial. A. 0; yes B. ; no C. 0; no D. ; yes. Solve x 0x 8 by completing the square.. A. 5 B. 5 C. 0 D Use synthetic division to divide x 5x 5x by x.. A. x 7x 9 R6 B. x x C. x D. x 7x 9 R6. Solve x.. A. B. C. D. 5. List the possible rational roots of x 5x x A.,,, B.,,, C.,, D.,, 6. Determine the rational roots of x x 6x A. B. C. D., 9 7. Solve x x x x 7 6 x. x 5x 6 A. B., C. 6 D., Find the discriminant of 5x 8x 0 and describe the 8. nature of the roots of the equation. A. ; two distinct real roots B. 0; exactly one real root C. 76; no real roots D. ; two distinct real roots 9. Solve x x 0 by using the Quadratic Formula. 9. A. i B. i C. i D., 6 0. Determine between which consecutive integers one or more 0. real zeros of ƒ(x) x x x 5 are located. A. 0 and B. and C. and D. at 5. Find the value of k so that the remainder of. (x x kx ) (x ) is 0. A. B. 0 C. D. 0 Glencoe/McGraw-Hill 57 Advanced Mathematical Concepts 38 Chapter Chapter Test, Form B (continued). Find the number of possible negative real zeros for. ƒ(x) x x x 9. A. or 0 B. or C. D. 0. Solve x 7.. A. x 79 B. x, x 79 C. x D. x 79. Decompose x into partial fractions. x 9x 5 A. x x B. 5 x x 5 C. x 5 x D. x x 5 5. Solve x. x x x x A. x B. x 0 C. x 0 D. x Approximate the real zeros of ƒ(x) x 5x to the nearest tenth. 6. A. 0.5, 0.7,.9 B. 0.6, 0.7,.9 C. 0.6 D. 0.6, 0.7, Solve 5 x 8 x A. 7 B. 0 C. 7 D. 8. Which polynomial function best models the set of data below? 8. x f(x) A. y 0.9x.9x 0.5x. B. y 0.9x.9x 0.5x. C. y.0x.9x 0.9x 7.5 D. y 0.9x.8x.x Solve x. x A. x 0 B. x, x C. 0 x D. 0 x, x Find the polynomial equation of least degree with roots,, and i. 0. A. x x 6x 9 0 B. x x 6x 8x 7 0 C. x x 6x 8x 7 0 D. x x x 8x 7 0 Bonus Solve x. Bonus: A., i B., C. D., i Glencoe/McGraw-Hill 58 Advanced Mathematical Concepts 39 Chapter Chapter Test, Form C Write the letter for the correct answer in the blank at the right of each problem.. Use the Remainder Theorem to nd the remainder when. x x x 7 is divided by x. State whether the binomial is a factor of the polynomial. A. 5; yes B. ; no C. ; no D. ; yes. Solve x 0x 575 by completing the square.. A. 5, 5 B C D. 5, 5. Use synthetic division to divide x x 5x by x.. A. x x 8 R7 B. x x R C. x x 8 R9 D. x x R5. Solve x.. A. 6 B. 6 C. 6 D List the possible rational roots of x x A.,,,,,, B.,,,, 6,,, C.,,, D.,,,, 6. Determine the rational roots of x 7x x A., B., C. D. 7. Solve. x x x x A. B. 6 C. 6 D Find the discriminant of 6x 9x 0 and describe the nature 8. of the roots of the equation. A. 9; two distinct real roots B. 0; exactly one real root C. 75; no real roots D. 75; two distinct real roots 9. Solve x x 7 0 by using the Quadratic Formula. 9. A. i0 B. i 0 C. i 0 D. 5i 0. Determine between which consecutive integers one or more real 0. zeros of ƒ(x) x x 5 are located. A. and B. and C. and D. and 0. Find the value of k so that the remainder of (x 5x x k). (x 5) is 0. A. 0 B. 0 C. 5 D. 0 Glencoe/McGraw-Hill 59 Advanced Mathematical Concepts Quick Reference ebook This le is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed More information 1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coe cients. Each of the separate parts, More information This is a square root. The number under the radical is 9. (An asterisk * means multiply.) Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize More information 40 Chapter Chapter Test, Form C (continued). Find the number of possible negative real zeros for. ƒ(x) x x x. A. B. or 0 C. or D.. Solve x.. A. x 79 B. x 79 C. x D. x 79 8x. Decompose into partial fractions. x x A. 6 x x B. x 6 x C. x 6 x D. x 6 x 5. Solve 8 x x x 0. x A. 9 x 0 B. x 0, x C. 9 x D. 9 x 0, x Approximate the real zeros of ƒ(x) x x to the nearest tenth. 6. A..0 B..0, 0.5 C..0, 0.0 D Solve 6x x. 7. A. B. C. D. 8. Which polynomial function best models the set of data below? 8. x f(x) A. y 0.x 0.x.x.0 B. y x 0x 7x 99 C. y 0.0x 0.0x.7x.99 D. y 0.0x 0.0x.7x Solve 5 x 7 6. A. x B. x C. x, x D. x Find a polynomial equation of least degree with 0. roots, 0, and. A. x x x 9 0 B. x x x 9 0 C. x 9x 0 D. x 9x 0 Bonus Solve 6x 6x x 6x 9 0. Bonus: A., B., C., D.,0 Glencoe/McGraw-Hill 60 Advanced Mathematical Concepts 41 Chapter Chapter Test, Form A Solve each equation or inequality.. (x ).. t 6t 5.. a 5.. x 5 x. 5. b 5b x x x x d 6 d Use the Remainder Theorem to nd the remainder when 8. x 5 x x is divided by x. State whether the binomial is a factor of the polynomial. 9. Determine between which consecutive integers the 9. real zeros of ƒ(x) x x 5x x 6 are located. 0. Decompose x x 9 into partial fractions. 5x 0.. Find the value of k so that the remainder of. (x x kx 0x ) (x ) is 0.. Approximate the real zeros of ƒ(x) x x 0 to. the nearest tenth. Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts 42 Chapter Chapter Test, Form A (continued). Use the Upper Bound Theorem to nd an integral upper. bound and the Lower Bound Theorem to nd an integral lower bound of the zeros of ƒ(x) x x.. Write a polynomial function with integral coe cients to. model the set of data below. x f( x) Find the discriminant of 5x x and describe the 5. nature of the roots of the equation. 6. Find the number of possible positive real zeros and the 6. number of possible negative real zeros for ƒ(x) x 7x 5x 8x. 7. List the possible rational roots of x x 7x Determine the rational roots of x 6x x Write a polynomial equation of least degree with 9. roots,, i, and i. How many times does the graph of the related function intersect the x-axis? 0. Francesca jumps upward on a trampoline with an initial 0. velocity of 7 feet per second. The distance d(t) traveled by a free-falling object can be modeled by the formula d(t) v 0 t gt, where v 0 is the initial velocity and g represents the acceleration due to gravity ( feet per second squared). Find the maximum height that Francesca will travel above the trampoline on this jump. Bonus Find ƒ if ƒ is a cubic polynomial function such Bonus: that ƒ(0) 0 and ƒ(x) is positive only when x. Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts 43 Chapter Chapter Test, Form B Solve each equation or inequality.. 8x 5x 0.. x x.. 6 q. q. 0x n a a a a a x Use the Remainder Theorem to nd the remainder when 8. x 5x 5x is divided by x. State whether the binomial is a factor of the polynomial. 9. Determine between which consecutive integers the real 9. zeros of ƒ(x) x x x are located. 0. Decompose 8x 7 into partial fractions. x x 0.. Find the value of k so that the remainder of. (x x kx 6) (x ) is 0. Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts 44 Chapter Chapter Test, Form B (continued). Approximate the real zeros of ƒ(x) x x to. the nearest tenth.. Use the Upper Bound Theorem to nd an integral upper. bound and the Lower Bound Theorem to nd an integral lower bound of the zeros of ƒ(x) x x.. Write a polynomial function to model the set of data below.. x f(x) Find the discriminant of x x 9 and describe 5. the nature of the roots of the equation. 6. Find the number of possible positive real zeros and 6. the number of possible negative real zeros for ƒ(x) x x x List the possible rational roots of x 5x x Determine the rational roots of x x 6x Write a polynomial equation of least degree with roots, 9.,, and. How many times does the graph of the related function intersect the x-axis? 0. Belinda is jumping on a trampoline. After jumps, she 0. jumps up with an initial velocity of 7 feet per second. The function d(t) 7t 6t gives the height in feet of Belinda above the trampoline as a function of time in seconds after the fth jump. How long after her fth jump will it take for her to return to the trampoline again? Bonus Factor x x x. Bonus: Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts 45 Chapter Chapter Test, Form C Solve each equation or inequality.. x x x 6x x. y. 5. x y y x Use the Remainder Theorem to nd the remainder when 8. x 5x 6x is divided by x. State whether the binomial is a factor of the polynomial. 9. Determine between which consecutive integers the 9. real zeros of ƒ(x) x x 5 are located. 0. Decompose 0x into partial fractions. x 0.. Find the value of k so that the remainder of. (x kx ) (x ) is zero.. Approximate the real zeros of ƒ(x) x x x 5 to. the nearest tenth. Glencoe/McGraw-Hill 65 Advanced Mathematical Concepts 46 Chapter Chapter Test, Form C (continued). Use the Upper Bound Theorem to SECTION 1-6 Quadratic Equations and Applications 58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be More information MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will More information 2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you rst learned nd an integral upper. bound and the Lower Bound Theorem to nd an integral lower bound of the zeros of ƒ(x) x x.. Write a polynomial function to model the set of data below.. x f(x) Find the discriminant of x x 7 0 and describe the 5. nature of the roots of the equation. 6. Find the number of possible positive real zeros and the 6. number of possible negative real zeros for ƒ(x) x x x. 7. List the possible rational roots of x x 7x Determine the rational roots of x x 7x Write a polynomial equation of least degree with 9. roots,, and 5. How many times does the graph of the related function intersect the x-axis? 0. What type of polynomial function could be the best model 0. for the set of data below? x 0 f(x) Bonus Determine the value of k such that Bonus: ƒ(x) kx x 7x 9 has possible rational roots of,, 9, 6,,,, 9. Glencoe/McGraw-Hill 66 Advanced Mathematical Concepts 47 Chapter Chapter Open-Ended Assessment Instructions: Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem.. Use what you have learned about the discriminant to answer the following. a. Write a polynomial equation with two imaginary roots. Explain your answer. b. Write a polynomial equation with two real roots. Explain your answer. c. Write a polynomial equation with one real root. Explain your answer.. Given the function ƒ(x) 6x 5 x 5x x x, answer the following. a. How many positive real zeros are possible? Explain. b. How many negative real zeros are possible? Explain. c. What are the possible rational zeros? Explain. d. Is it possible that there are no real zeros? Explain. e. Are there any real zeros greater than? Explain. f. What term could you add to the above polynomial to increase the number of possible positive real zeros by one? Does the term you added increase the number of possible negative real zeros? How do you know? g. Write a polynomial equation. Then describe its roots.. A 6-foot-tall light pole has a 9-foot-long wire attached to its top. A stake will be driven into the ground to secure the other end of the wire. The distance from the pole to where the stake should be driven is given by the equation 9 d 6, where d represents the distance. a. Find d. b. What relationship was used to write the given equation? What do the values 9, 6, and d represent? Glencoe/McGraw-Hill 67 Advanced Mathematical Concepts 48 Chapter Chapter, Mid-Chapter Test (Lessons - through -). Determine whether is a root of x x 5x 0.. Explain.. Write a polynomial equation of least degree with roots.,, i, and i. How many times does the graph of the related function intersect the x-axis?. Find the complex roots for the equation x Solve the equation x 6x 0 0 by completing. the square. 5. Find the discriminant of 6 5x 6x. Then solve the 5. equation by using the Quadratic Formula. 6. Use the Remainder Theorem to nd the remainder when 6. x x is divided by x. State whether the binomial is a factor of the polynomial. 7. Find the value of k so that the remainder of 7. (x 5x kx ) (x ) is 0. List the possible rational roots of each equation. Then determine the rational roots. 8. x 0x x 7x Find the number of possible positive real zeros and the 0. number of possible negative real zeros for the function ƒ(x) x x x. Then determine the rational zeros. Glencoe/McGraw-Hill 68 Advanced Mathematical Concepts 49 Chapter Chapter, Quiz A (Lessons - and -). Determine whether is a root of x x x 0.. Explain.. Write a polynomial equation of least degree with. roots,, i, and i. How many times does the graph of the related function intersect the x-axis?. Find the complex roots of the equation x x 0... Solve x 0x 5 0 by completing the square.. 5. Find the discriminant of 5x x and describe 5. the nature of the roots of the equation. Then solve the equation by using the Quadratic Formula. Chapter Chapter, Quiz B (Lessons - and -) Use the Remainder Theorem to nd the remainder for each division. State whether the binomial is a factor of the polynomial.. (x 6x 9) (x ).. (x 6x 8) (x ).. Find the value of k so that the remainder of. (x 5x kx ) (x ) is 0.. List the possible rational roots of x x 5x 0.. Then determine the rational roots. 5. Find the number of possible positive real zeros and the 5. number of possible negative real zeros for ƒ(x) x 9x x. Then determine the rational zeros. Glencoe/McGraw-Hill 69 Advanced Mathematical Concepts More information Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative More information Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks More information 50 Chapter Chapter, Quiz C (Lessons -5 and -6). Determine between which consecutive integers the real. zeros of ƒ(x) x x x 5 are located.. Approximate the real zeros of ƒ(x) x x x to. the nearest tenth. a. Solve a 6 a... Decompose p p p into partial fractions. p p. 5. Solve w w Chapter Chapter, Quiz D (Lessons -7 and -8) Solve each equation or inequality.. x.. m.. t Determine the type of polynomial. function that would best t the scatter plot shown. 5. Write a polynomial function with integral coe cients to 5. model the set of data below. x f(x) Glencoe/McGraw-Hill 70 Advanced Mathematical Concepts 51 Chapter Chapter SAT and ACT Practice After working each problem, record the correct answer on the answer sheet provided or use your own paper. Multiple Choice. The vertices of a parallelogram are P(0, ), Q(, 0), R(7, ), and S(, 6). Find the length of the longer sides. A B C 7 D 5 E None of these. A right triangle has vertices A(5, 5), B(5 x, 9), and C(, 9). Find the value of x. A 0 B 9 C D 0 E A B 7 C 8 D E A 5 55 B ( 55) C 55 D E None of these 5. Find the slope of a line perpendicular to x y 7. A B C D E 7 6. If the midpoint of the segment joining points A, 5 and B x, 5 has coordinates 5,, nd the value of x. A B C D E None of these 7. If 6 x 9, then 8 x A B C D E 6 8. If a pounds of potatoes serves b adults, how many adults can be served with c pounds of potatoes? A a c b B b c a C b a c D c a b E It cannot be determined from the information given. 9. Which are the coordinates of P, Q, R, and S if lines PQ and RS are neither parallel nor perpendicular? A P(, ), Q(, ), R(0, 5), S(, ) B P(6, 0), Q(, 0), R(, 8), S(, 5) C P(5, ), Q(7, ), R(, ), S(, ) D P(8, ), Q(, 0), R(, 5), S(6, ) E P(6, 8), Q(0, 6), R(, 5), S(7, ) 0. What is the length of the line segment whose endpoints are represented by the points C(6, 9) and D(8, )? A 58 B 0 C 7 D 58 E 0 Glencoe/McGraw-Hill 7 Advanced Mathematical Concepts 52 Chapter Chapter SAT and ACT Practice. If x# means (x ), nd the value of (#)#. A 8 B 0 C D 6 E 6. Each number below is the product of two consecutive positive integers. For which of these is the greater of the two consecutive integers an even integer? A 6 B 0 C D 56 E 7. For which equation is the sum of the roots the greatest? A (x 6) B (x ) 9 C (x 5) 6 D (x 8) 5 E x 6 6. Find the midpoint of the segment with endpoints at (a b, c) and (a, c). A a, b c B a b,c C a, c D (c, b a) E (a b, c) 7 8. Quantitative Comparison A if the quantity in Column A is greater B if the quantity in Column B is greater C if the two quantities are equal D if the relationship cannot be determined from the information given Column A 7. 0 y x Column B x y x y. If a a, then a 8. x A 6 C B D 5 x 6x 9 x x 9 x E 5. For which value of k are the points A(0, 5), B(6, k), and C(, ) collinear? A 7 B C D E 9. Grid-In If the slope of line AB is and lines AB and CD are parallel, what is the value of x if the coordinates of C and D are (0, ) and (x, ), respectively? 0. Grid-In A parallelogram has vertices at A(, ), B(, 5), C(, ), and D(, 0). What is the x-coordinate of the point at which the diagonals bisect each other? Glencoe/McGraw-Hill 7 Advanced Mathematical Concepts 53 Chapter Chapter Cumulative Review (Chapters -). State the domain and range of the relation {(, 5), (, ),. (0, 5)}. Then state whether the relation is a function. Write yes or no.. Find [ƒ g](x) if ƒ(x) x 5 and g(x) x... Graph y x... Solve the system of equations. x y z 0. x y z 6 x y z 5. The coordinates of the vertices of ABC are A(, ), 5. B(, ), and C(, ). Find the coordinates of the vertices of the image of ABC after a 70 counterclockwise rotation about the origin. 6. Gabriel works no more than 5 hours per week during the 6. school year. He is paid $ per hour for tutoring math and $9 per hour for working at the grocery store. He does not want to tutor for more than 8 hours per week. What are Gabriel s maximum earnings? 7. Determine whether the graph of y x is symmetric to the x-axis, the y-axis, the line y x, the line y x, or none of these Graph y x using the graph of the function y x Describe the end behavior of y x 7 x Determine the slant asymptote for ƒ(x) x x. x. Solve x x x x 8 x Solve x x x 0 x.. Glencoe/McGraw-Hill 7 Advanced Mathematical Concepts Algebra Unpacked Content For the new Common Core standards that will be e ective in all North This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI sta are continually updating and improving these tools to better serve teachers. Algebra More information of surface, 569-571, 576-577, 578581 of triangle, 548 Associative Property of addition, 12, 331 of Absolute Value and arithmetic, 730-733 de ned, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property More information 54 BLANK 55 UNIT Unit Review, Chapters Given that x is an integer, state the relation representing each equation as a set of ordered pairs. Then, state whether the relation is a function. Write yes or no.. y x and x. y x and x Find [ƒ g](x) and [g ƒ](x) for each ƒ(x) and g(x).. ƒ(x) x g(x) x. ƒ(x) x g(x) x 5. ƒ(x) x 5 g(x) x Find the zero of each function. 6. ƒ(x) x 0 7. ƒ(x) 5x 8. ƒ(x) 0.75x Write the slope-intercept form of the equation of the line through the points with the given coordinates. 9. (, ), (6, 0) 0. (, ), (5, ) Write the standard form of the equation of each line described below.. parallel to y x passes through (, ). perpendicular to x y 6 x-intercept: The table below shows the number of T-shirts sold per day during the rst week of a senior-class fund-raiser. Day Number of Shirts Sold Use the ordered pairs (, ) and (, ) to write the equation of a best- t line.. Predict the number of shirts sold on the eighth day of the fund-raiser. Explain whether you think the prediction is reliable. Graph each function. 5. ƒ(x) x 6. ƒ(x) x x if x 7. ƒ(x) x if x x if x Graph each inequality. 8. x y 9. y x 5 Solve each system of equations. 0. y x x y 5. x y x y. 7x z y z 8 x y 7 Use matrices A, B, C, and D to nd each sum, di erence, or product. 6 A B C A B. A B 5. CD 6. AB CD D E Use matrices A, B, and E above to nd the following. 7. Evaluate the determinant of matrix A. 8. Evaluate the determinant of matrix E. 9. Find the inverse of matrix B. Solve each system of inequalities by graphing. Name the coordinates of the vertices of each polygonal convex set. Then, nd the maximum and minimum values for the function ƒ(x, y) y x. 0. x 0. x y 0 y x y y 5 x y x 8 Glencoe/McGraw-Hill 75 Advanced Mathematical Concepts 56 UNIT Unit Review, Chapters (continued) Determine whether each function is an even function, an odd function, or neither.. y x. y x 5. y x x 6x 8 Use the graph of ƒ(x) x to sketch a graph for each function. Then, describe the transformations that have taken place in the related graphs. 5. y ƒ(x) 6. y ƒ(x ) Graph each inequality. 7. y x 8. y x Find the inverse of each function. Sketch the function and its inverse. Is the inverse a function? Write yes or no. 9. y x 5 0. y (x ) Determine whether each graph has in nite discontinuity, jump discontinuity, point discontinuity, or is continuous. Then, graph each function.. y x x x if x 0. y x if x 0 Find the critical points for the functions graphed in Exercises and. Then, determine whether each point is a maximum, a minimum, or a point of in ection... Determine any horizontal, vertical, or slant asymptotes or point discontinuity in the graph of each function. Then, graph each function. 5. y x (x )(x ) 6. y x 9 x Solve each equation or inequality. 7. x 8x x x x x x ; x x 5. 9 x 5. x 8 x 5 Use the Remainder Theorem to nd the remainder for each division. 5. (x x ) (x 6) 5. (x x ) (x ) Find the number of possible positive real zeros and the number of possible negative real zeros. Determine all of the rational zeros. 55. ƒ(x) x x 56. ƒ(x) x x x x Approximate the real zeros of each function to the nearest tenth. 57. ƒ(x) x x ƒ(x) x x x Glencoe/McGraw-Hill 76 Advanced Mathematical Concepts 57 UNIT Unit Test, Chapters. Find the maximum and minimum values of ƒ(x, y) x y. for the polygonal convex set determined by x, y 0, and x 0.5y.. Write the polynomial equation of least degree that has the. roots i,i, i, and i.. Divide x x x 75 by x by using synthetic. division.. Solve the system of equations by graphing.. x 5y 8 x y 5. Complete the graph so that it is the graph of an 5. even function. 6. Solve the system of equations. 6. x y z x y z x y z 5 7. Decompose the expression 7n into partial fractions. 7. n n 6 8. Is the graph of x y 9 symmetric with respect to the 5 x-axis, the y-axis, neither axis, or both axes? Without graphing, describe the end behavior of the graph 9. of ƒ(x) 5x x. 0. How many solutions does a consistent and dependent 0. system of linear equations have?. Solve x 7x Solve y y 0.. Glencoe/McGraw-Hill 77 Advanced Mathematical Concepts 58 UNIT Unit Test, Chapters (continued). If ƒ(x) x and g(x) x, nd [ g ƒ](x)... Are ƒ(x) x 5 and g(x) x 5 inverses of each other?. 5. Find the inverse of y x. Then, state whether the 0 inverse is a function Determine if the expression m 5 6m 8 m is a 6. polynomial in one variable. If so, state the degree. 7. Describe how the graph of y x is related to 7. its parent graph. 8. Write the slope-intercept form of the equation of the line 8. that passes through the point (5, ) and has a slope of. 9. Determine whether the gure with vertices at 9. (, ), (, ), (, ), and (, ) is a parallelogram. 0. A plane ies with a ground speed of 60 miles per hour 0. if there is no wind. It travels 50 miles with a head wind in the same time it takes to go 50 miles with a tail wind. Find the speed of the wind.. Solve the system of equations algebraically.. x y x y 9 5. Find the value of 0 by using expansion by minors... Solve the system of equations by using augmented matrices.. y x 0 x y. Approximate the greatest real zero of the function. g(x) x x to the nearest tenth. 5. Graph ƒ(x) x. 5. Glencoe/McGraw-Hill 78 Advanced Mathematical Concepts 6.1 Add & Subtract Polynomial Expression & Functions 6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coe cient, like terms, polynomial funciton, quardrtic More information Review of Intermediate Algebra Content Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6 More information Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coe cients and p/q (where p/q is reduced) is a rational zero, then p is a factor of 59 UNIT Unit Test, Chapters (continued) 6. Write the slope-intercept form 6. of the equation 6x y 9 0. Then, graph the equation. 7. Write the standard form of the equation of the line 7. that passes through (, 7) and is perpendicular to the line with equation y x Use the Remainder Theorem to nd the remainder of 8. (x 5x 7x ) (x ). State whether the binomial is a factor of the polynomial. 9. Solve x x Determine the value of w so that the line whose equation 0. is 5x y w passes through the point at (, ).. Determine the slant asymptote for ƒ(x) x 5. x x Find the value of... State the domain and range of {(5, ), (, ), (, 0), (5, )}.. Then, state whether the relation is a function.. Determine whether the function ƒ(x) x is odd,. even, or neither. 5. Find the least integral upper bound of the zeros of the 5. function ƒ(x) x x. 6. Solve x If A and B 0, nd AB Name all the values of x that are not in the domain of 8. ƒ(x). x x 5 9. Given that x is an integer between and, state the 9. relation represented by the equation y x by listing a set of ordered pairs. Then, state whether the relation is a function. Write yes or no. Glencoe/McGraw-Hill 79 Advanced Mathematical Concepts 60 UNIT Unit Test, Chapters (continued) 0. Determine whether the system of 0. inequalities graphed at the right is infeasible, has alternate optimal solutions, or is unbounded for the function ƒ(x, y) x y.. Solve ( y )(y )... Determine whether the function y x has in nite discontinuity, jump discontinuity, or point discontinuity, or is continuous... Find the slope of the line passing through the points. at (a, a ) and (a, a 5).. Together, two printers can print 7500 lines if the rst. printer prints for minutes and the second prints for minute. If the rst printer prints for minute and the second printer prints for minutes, they can print 9000 lines together. Find the number of lines per minute that each printer prints. 5. A box for shipping roo ng nails must have a volume of 5. 8 cubic feet. If the box must be feet wide and its height must be feet less than its length, what should the dimensions of the box be? 6. Solve the system of equations. 6. x y z x 5y z 6 x y z 7. Solve x x 7 0 by completing the square Find the critical point of the function y (x ). 8. Then, determine whether the point represents a maximum, a minimum, or a point of in ection. x y 5 9. Solve Write the standard form of the equation of the line 50. that passes through (5, ) and is parallel to the line with equation x y 0. Glencoe/McGraw-Hill 80 Advanced Mathematical Concepts More information Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson More information 61 Glencoe/McGraw-Hill A Advanced Mathematical Concepts SAT and ACT Practice Answer Sheet (0 Questions) / / CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide 62 Glencoe/McGraw-Hill A Advanced Mathematical Concepts SAT and ACT Practice Answer Sheet (0 Questions) / / / / View more MATH 21. College Algebra 1 Lecture Notes MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a More information Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coe cient Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials Scope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is rst introduced More information More information Polynomial Degree and Finite Di erences CONDENSED LESSON 7.1 Polynomial Degree and Finite Di erences In this lesson you will learn the terminology associated with polynomials use the nite di erences method to determine the degree of a polynomial More information Factoring Polynomials UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can More information Polynomial Expressions and Equations Polynomial Expressions and Equations This is a really close-up picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen More information Algebra 1. Curriculum Map Algebra 1 Curriculum Map Table of Contents Unit 1: Expressions and Unit 2: Linear Unit 3: Representing Linear Unit 4: Linear Inequalities Unit 5: Systems of Linear Unit 6: Polynomials Unit 7: Factoring More information Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic More information 3.3 Real Zeros of Polynomials 3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section More information Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation More information Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x _.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic More information SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra More information 7.1 Graphs of Quadratic Functions in Vertex Form 7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called More information 2.4 Real Zeros of Polynomial Functions SECTION 2.4 Real Zeros of Polynomial Functions 197 What you ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower More information JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials More information Students will be able to simplify and evaluate numerical and variable expressions using Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable More information 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise More information Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important More information More information Copyrighted Material. Chapter 1 DEGREE OF A CURVE Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and o er two More information More information The program also provides supplemental modules on topics in geometry and probability and Algebra 1 Course Overview Students develop algebraic uency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students More information Algebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scienti c notation to express More information More information How do you compare numbers? On a number line, larger numbers are to the right and smaller The verbal answers to all of the following questions should be memorized before completion of prealgebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics More information Warm-Up Oct. 22. Daily Agenda: Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment More information Zeros of Polynomial Functions Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate More information What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything More information CAMI Education linked to CAPS: Mathematics - 1 - TOPIC 1.1 Whole numbers _CAPS curriculum TERM 1 CONTENT Mental calculations Revise: Multiplication of whole numbers to at least 12 12 Ordering and comparing whole numbers Revise prime numbers to More information Section 5.0A Factoring Part 1 Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the nal result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) ( More information More information ALGEBRA I (Created 2014) Amherst County Public Schools Prentice Hall Mathematics, Algebra 1 2009 ALGEBRA I (Created 2014) Amherst County Public Schools The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and ampli es Prentice Hall Mathematics, Algebra 1 2009 Grades 9-12 C O R R E L A T E D T O Grades 9-12 Prentice Hall Mathematics, Algebra 1 Program Organization Prentice Hall Mathematics supports student comprehension More information South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Indiana State Core Curriculum Standards updated 2009 Algebra I South Carolina College- and CareerReady (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language, Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through Week 13 Trigonometric Form of Complex Numbers In mathematics, there are four attainment targets: using and applying mathematics; number and Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scienti c notation to express large numbers and small More information More information Review of Fundamental Mathematics Algebra 1 Course Information More information ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n +an1xn1++a1x Course Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Pre-Algebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and More information a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4) More information UNCORRECTED PAGE PROOFS number and and algebra TopIC 17 Polynomials 17.1 Overview Why learn this? Just as number is learned in stages, so too are graphs. You have been building your knowledge of graphs and functions over time. More information More information Mathematics Pre-Test Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11} Mathematics Pre-Test Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {-1, 1} III. {-1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following More information A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scienti c notation to represent real numbers. Use properties of radicals. Simplify More information Polynomial Operations and Factoring Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coe cients, and degree of polynomials. More information McDougal Littell California: McDougal Littell California: Pre-Algebra Algebra 1 correlated to the California Math Content s Grades 7 8 McDougal Littell California Pre-Algebra Components: Pupil Edition (PE), Teacher s Edition (TE), More information SPECIAL PRODUCTS AND FACTORS CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Di erence of Two Terms 11-5 Factoring the More information Chapter 7 - Roots, Radicals, and Complex Numbers Math 233 - Spring 2009 Chapter 7 Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the More information Common Core Unit Summary Grades 6 to 8 Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity8G1-8G5 rotations re ections and translations,( RRT=congruence) understand congruence of 2 d gures after RRT Dilations More information Lesson 9: Radicals and Conjugates Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form. More information Mathematics Placement Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four di erent levels of math including pre-algebra, algebra, college algebra, and trigonometry. More information 2022 © DocPlayer.net Privacy Policy | Terms of Service | Feedback | Do Not Sell My Personal Information Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are More information Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts More information A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25 More information March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial More information Lagrange Interpolation is a method of tting an equation to a set of points that functions well when Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of tting an equation to a set of points that functions well when there are few points given. Sasha s method More information http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304 MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio More information Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Algebra 1 2008 STANDARDS PERFORMANCE STANDARDS A1:1 Number Sense.1 Classify numbers as Real, Irrational, Rational, Integer, More information Algebra I. In this technological age, mathematics is more important than ever. When students In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment, More information SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E) More information YOU CAN COUNT ON NUMBER LINES Key Idea 2 Number and Numeration: Students use number sense and numeration to develop an understanding of multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and More information The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method More information KEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 4-47) Surveys and More information 12) 13) 14) (5x)2/3. 16) x5/8 x3/8. 19) (r1/7 s1/7) 2 DMA 080 WORKSHEET # (8.-8.2) Name Find the square root. Assume that all variables represent positive real numbers. ) 6 2) 8 / 2) 9x8 ) -00 ) 8 27 2/ Use a calculator to approximate the square root to decimal More information This unit will lay the groundwork for later units where the students will extend this knowledge to Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course More information Algebra I Credit Recovery Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics, More information