10 Mathematics Quarter 2 – Module 1: Polynomial Functions The following are some reminders in using this module: 1. Read the instruction carefully before doing each task. 2. Observe honesty and integrity in doing the tasks and checking your answers. 3. Finish the task at hand before proceeding to the next. 4. For the submission of module, write all your answers in a word file and save it as a PDF with a file name of the given lesson. After that, send your file/s via messenger. Rolino G. Alvarez (Online Class) If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. Fighting!!! What I Need to Know At the end of the lesson, with at least 80% level of proficiency, students shouldbe able to: 1. illustrate Polynomial Functions; 2. understand, describe, and interpret the graphs of polynomial function (M10AL-IIa-1); and 3. solve problems involving polynomial functions (M10AL-IIb-2). Lesson Polynomial Functions 1 This module will be assessing your knowledge of the different math concept/s previously studied and your skill/s in performing mathematical operations. These knowledge and skills will help you understand quadratic inequalities. What’s In Identify if the given is a polynomial expression or not. Write P if it is considered as a polynomial expression and PN if not and give your reasons. 1. 6. 2. 7. √ 3. 4. √ √ 8. ⁄ 9. ⁄ 5. 10. What’s New Study the given polynomial functions and complete the table below. Polynomial Function 1. ( ) 2. ( ) 3. ( ) 4. ( ) 5. ( ) ( ) Polynomial Function in Standard Form Degree Leading Coefficient Constant Term What is It A polynomial function is a function of the form , ( ) where is a non-negative integer, the leading term, are real numbers called coefficients, is the leading coefficient, and is is the constant term. The terms of a polynomial may be written in any order. However, if they are written in decreasing powers of , we say the polynomial function is in standard form. Other than ( ), a polynomial function may also be denoted by polynomial function is represented by a set of ordered pairs ( ( ). Sometimes, a ). Thus, a polynomial function can be written in different ways, like the following. ( ) or Polynomials may also be written in factored form and as a product of irreducible factors, that is, a factor that can no longer be factored using coefficients that are real numbers. Here are some examples. a. in factored form is ( b. in factored form is ( c. in factored form is ( d. ( ) e. ( ) in factored form is ( ) )( ( in factored form is ( ) )( ( )( )( )( )( )( ) )( ) ) ) ) Factor each polynomial completely using any method. 1. ( 2. ( 3. ( )( ) )( ) )( ) 4. 5. The preceding task is very important for you since it has something to do with the x- intercepts of a graph. These are the x-values when y = 0, thus, the point(s) where the graph intersects the x-axis can be determined. To recall the relationship between factors and x-intercepts, consider these examples: Example 1: Find the intercepts of . Solution: To find the x-intercept/s, set . Use the factored form. That is, ( )( )( )( ) Factor completely. ( )( )( )( ) Equate to 0. Equate each factor to 0 to determine . The x-intercepts are -1, -2, 2 and 3. This means the graph will pass through ( ( ) and ( ). Finding the y-intercept is more straightforward. Simply set ), ( ), in the given polynomial. That is, ( ) ( ) ( ) The y-intercepts is 12. This means the graph will also pass through ( ). Example 2: Find the intercepts of . Solution: To find the x-intercept/s, set . Use the factored form. That is, ( )( )( ) Factor completely. ( )( )( ) Equate to 0. Equate each factor to 0 to determine . The x-intercepts are -2, -1, and 1. This means the graph will pass through ( and ( ). ), ( ), Again, finding the y-intercept simply requires us to set ( ) in the given polynomial. That is, ( ) The y-intercepts is -2. This means the graph will also pass through ( ). You have learned how to find the intercepts of a polynomial function. You will discover more properties as you go through the next activities. What’s More Determine the x-intercept/s and the y-intercept of each given polynomial function. To obtain other points on the graph, find the value of 1. ( )( -5 )( that corresponds to each value of in the table. ) -3 0 2 4 x-intercepts: _____, _____, _____, _____ y-intercepts: _____ List all your answers above as ordered pairs. 2. ( )( )( -6 ) -4 0 3 5 x-intercepts: _____, _____, _____ y-intercepts: _____ List all your answers above as ordered pairs. 3. ( )( -7 ) -3 1 2 x-intercepts: _____, _____, _____ y-intercepts: _____ List all your answers above as ordered pairs. 4. ( )( -4 )( -2 )( -0.5 ) 2 4 x-intercepts: _____, _____, _____, _____ y-intercepts: _____ List all your answers above as ordered pairs. What I Have Learned Study each figure and answer the questions that follow. Summarize your answers using a table similar to the one provided. Case 1 The graph on the right is defined by , or in factored form, ( )( )( ). Questions: a. Is the leading coefficient a positive or a negative number? b. Is the polynomial of even degree or odd degree? c. Observe the end behaviors of the graph on both sides. Is it rising or falling to the left or to the right? Case 2 The graph on the right is defined by , or in factored form, ( ) ( )( ) . Questions: a. Is the leading coefficient a positive or a negative number? b. Is the polynomial of even degree or odd degree? c. Observe the end behaviors of the graph on both sides. Is it rising or falling to the left or to the right? Case 3 The graph on the right is defined by , or in factored form, ( )( )( ). Questions: a. Is the leading coefficient a positive or a negative number? b. Is the polynomial of even degree or odd degree? c. Observe the end behaviors of the graph on both sides. Is it rising or falling to the left or to the right? Case 4 The graph on the right is defined by , or in factored form, ( )( )( )( ). Questions: a. Is the leading coefficient a positive or a negative number? b. Is the polynomial of even degree or odd degree? c. Observe the end behaviors of the graph on both sides. Is it rising or falling to the left or to the right? What I Can Do Complete the table below. In the last column, draw a possible graph for the function, showing how the function behaves. You do not need to place your graph on the - plane. The first one is done for you as an example. Behavior of Leading Sample Polynomial Function 1. 2. 3. 4. Coefficient: or Degree: Even or Odd Odd the Graph: Rising or Possible Falling Sketch Left Right hand hand falling rising Republic of the Philippines SOUTHERN LEYTE STATE UNIVERSITY – TOMAS OPPUS Junior Laboratory High School San Isidro, Tomas Oppus, Southern Leyte Activity in Polynomial Function A. Directions. Read each item carefully and write the letter of the correct answer. 1. Which of the following is a polynomial? 3 2 i. x 2 x 5 x 2 a. i only 3 2 ii. 5 x 3x x 2 b. ii only c. i and ii 5 x 2 3x iii. d. i and iii 2. The following are examples of polynomial, EXCEPT a. y2 4 y 5 4 3 c. 3r 5r 2r 1 b. 5x 3 9 x 2 12 x 8 3 3 d. a b 8 9 5 3 3. What is the leading coefficient of the polynomials, 4 x 5 x 4 x x x ? a. 4 b. 5 c. 8 d. 1 4. If you will be asked to choose from -2, 2, 3, and 4, what values for and will you consider so that could define the graph on the right? a. , b. , c. , d. , 5. Your friend Joan asks your help on drawing a rough sketch of the graph of ) by means of the leading coefficient test. How will you explain the behavior of )( the graph? a. The graph is falling to the left and rising to the right. b. The graph is rising to both left and right. c. The graph is rising to the left and falling to the right. d. The graph is falling to both left and right. 6. ( What is the degree of the polynomial function a. 1 b. 2 ? c. 3 d. 10 7. What are the end behaviors of the graph of ( ) ? a. rises both directions c. rises to the left and falls to the right b. falling both directions d. falls to the left 8. If you will draw the graph of ) , how will you illustrate it with respect to the x- ( axis? a. Illustrate it crossing both ( ) and ( b. Illustrate it tangent at both ( c. Illustrate it crossing ( d. Illustrate it tangent at ( 9. A polynomial graph of ). ) and ( ). ) and tangent at ( ). ) and crossing ( ). with real coefficients and degree 2 has an imaginary zero/root at has a -intercept at ( . The ). Find . a. ( ) c. ( ) b. ( ) d. ( ) 10. What are the -intercepts of a polynomial functions? a. ( b. Absolute ) c. value of d. roots B. Directions. Determine the intercepts of the following polynomial functions. (Provide a neat and clean solution). 1. 2. ( )( 3. ( )( 4. 5. ) )( ) Republic of the Philippines SOUTHERN LEYTE STATE UNIVERSITY – TOMAS OPPUS Junior Laboratory High School San Isidro, Tomas Oppus, Southern Leyte Reflection Paper Question: Write down 3 things you have learned about polynomial functions. What particular part of the lesson you find it difficult? What question/s you can generate out from the lesson. ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ _____________________________________________________________________________________________________________