Name: Date: Period: Polynomials Lessons: 6-1, 6-4, 6-5 Packet 7 Tennessee State Standards 3103.3.2 Determine roots of a higher order polynomial. Common Core State Standards A-SSE- 2 Use the structure of an expression to identify ways to rewrite it. For example, a difference of two squares A-APR-2 Know and apply the Remainder Theorem: For a Polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). A-APR-3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Name: Lesson 6-1 Date: Period: Polynomials p.1 Degree Leading Coefficient Polynomial Terms Degree Leading Coefficient Binomial Trinomial Polynomial Classifying Polynomials by Degree Name: Date: Lesson 6-1 Polynomials Period: p.2 Classifying Polynomials 2x + 4x3 – 1 1. Write terms in descending order (based on degree). 2. Identify the Leading Coefficient 3. Identify the degree 4. How many terms? 5. Classify/give a name Part A: Guided Practice 1. Write terms in descending order (based on degree). 2. Identify the Leading Coefficient 3. Identify the degree 4. How many terms? 5. Classify/give a name 4x + 2x2 – 7 + x3 5x2 – 4x3 4x4 + 8x2 + 1 – 3x Name: Date: Lesson 6-4 Period: Factoring Polynomials p.3 Determining Whether a Binomial is a Factor of a Polynomial 1. To determine if (x – 3 ) is a factor, use substitution. If f(3) = 0, (x – 3) IS a factor. If f(3) = anything else, it is not. Is (x – 3) a factor of f(x)= x2 + 2x + 3? 2. Use a calculator to substitute 3 for x in f(x). 3. What is the result? Is (x – 3) a factor? 1. If f(-4)= 0, (x + 4) is a factor. If f(4) = anything else, it is not. 2. Substitute -4 for x in f(x). **If using a calculator, be sure to put -4 in parentheses (-4) each time! Why is this important? 3. What is the result? Is (x + 4) a factor? Is (x+4) a factor of f(x) = 2x4 + 8x3 + 2x + 8? Name: Date: Lesson 6-4 Factoring Polynomials Period: p.4 Review: Factoring a Difference of Two Squares 1. Recognize that each of the two terms is a perfect square. Coefficients are perfect squares and exponents are all even. There is a minus sign (-) in between, making it a difference. 25x4 - 81 2. Take the square root of the first term. 3. Take the square root of the second term. 4. One factor is (a + b), the other is (a – b).**These are called conjugates. Other Popular Factoring Methods Name What it looks like Method Name: Date: Lesson 6-4, 6-5 Factoring Polynomials Period: p.5 Part A: Guided Practice Factor. 1. 16- 100x6 2. 5x4 + 40x 3. 8x3 – 27 Finding the Roots of a Polynomial Equation *A root is another name for a ________________________. 1. Factor out a GCF, if possible. 2. Factor the trinomial or binomial. 3. Set each factor = 0. 4. Solve for x. 5. These are the roots. 6. Check using the calculator. - Graph the function - Calc (2nd Trace) - #2 Zero - Left Bound, Right Bound, Guess 2x3 + 4x2 – 30x Name: Lesson 6-5 Date: Roots of Polynomials Period: p.6 Part A: Guided Practice Find the roots (solve). Find the roots (solve). 3x4 – 6x2 – 24 X4 – 13x2 = -36 Name: Date: Lesson 6-5 Roots of Polynomials Period: p. 7 Multiplicity Multiplicity is the _________ of times a _______ occurs as a factor. Even multiplicity means that the graph ____________ the x-axis. Odd multiplicity means that the graph ____________ the x-axis. 1. Look at the graph of x3 – 9x2 + 27x – 27. The root is 3. What does the graph do at x = 3? _____________ Draw a rough sketch of the graph. 2. Factor to find the roots of this polynomial. State what the graph does at each of those roots based on the multiplicity. f(x) = 4x6 + 4x5 – 24x4 Name: Date: Lesson 6-5 Roots of Polynomials Period: p. 8 The Rational Root Theorem This theorem helps you list all the 4x4 – 21x3 + 18x2 + 19x – 6 = 0 POSSIBLE factors of a polynomial. 1. Write all the factors of the constant term. (positive and negative) 2. Write all the factors of the leading coefficient. (positive and negative) 3. Divide every number on the first list by every number on the second list. 4. Simplify each number. 5. Cross out duplicates. 6. These are the possible roots! Name: Lesson 6-5 Date: Roots of Polynomials Period: p. 9 *Now we will use our list and find out which possible roots really are roots. 1. Copy down the list of roots from the previous page. 2. Write down the polynomial. 3. Begin using substitution to test each of the roots. When one results in 0, it is a root! 4. To check, graph the polynomial! Name: Lesson 6-5 Date: Roots of Polynomials Period: p.10 Part B: Homework 1. Identify the leading coefficient, degree, and number of terms. Name the polynomial. 5x2 – 4x3 2. Verify whether or not (x – 2) is a factor of x2 + 3x – 7. Show your work. 3. Verify whether or not (x – 8) is a factor of x5 – 8x4 + 8x – 64. Show your work. 4. Factor x6 – 1 5. Factor 40 – 5x3 6. List all the possible roots x4 + 3x3 – 3x2 – 12x – 4 7. List all the possible roots x3 + x2 – 2x – 8 = 0 8. Find all the real roots by factoring. State the multiplicity of each. x5 – 2x4 – 24x3 = 0 9. Find all the real roots by factoring. State the multiplicity of each. 3x5 + 18x4 – 21x3 = 0 10. List all the possible roots AND test each one using substitution. Show your work. x4 – 3x2 – 4 = 0 Show your work for #10 here.