Name Class Date. Finding Real Roots of Polynomial Equations Extension: Graphing Factorable Polynomial Functions

Transcription

1 Name Class Date -1 Finding Real Roots of Polnomial Equations Etension: Graphing Factorable Polnomial Functions Essential question: How do ou use zeros to graph polnomial functions? Video Tutor prep for MCC9 1.F.IF.7c 1 Engage Understanding Polnomial Functions A polnomial function can be written as f () = a n n + a n - 1 n a 1 + a 0 where the coefficients a n,..., a 1, a 0 are real numbers. This is known as standard form. Note that linear functions are polnomial functions of degree 1 and quadratic functions are polnomial functions of degree. The functions ou have graphed in the form f () = a( - h) n + k are also polnomial functions, as ou will see in later lessons. The end behavior of a polnomial function f() = a n n + a n - 1 n a 1 + a 0 is determined b the term with the greatest degree, a n n. Polnomial End Behavior n a n As + As - Graph Even Positive f() + f() + Houghton Mifflin Harcourt Publishing Compan Even Negative f() - f() - Odd Positive f() + f() - Odd Negative f() - f() + You know how to use transformations to help ou graph functions of the form f () = a ( - h) n + k. If a polnomial function is not written in this form, then other graphing methods must be used. You will learn several methods throughout this unit, but the most basic is to plot points and connect them with a smooth curve, taking into account the end behavior of the function. Module 109 Lesson 1

2 In general, the higher the degree of a polnomial, the more comple the graph. Here are some tpical graphs for polnomials of degree n. Degree 1 Linear Degree Quadratic Degree 3 Cubic Degree Quartic REFLECT 1a. Evaluate each term of the polnomial f () = for = 10 and = -10. Eplain wh the end behavior is determined b the term with the greatest degree. 1b. Describe the end behavior of f () = Eplain our reasoning. The nested form of a polnomial has the form f () = (((a + b) + c) + d)...). It is often simpler to evaluate polnomials in nested form. MCC9 1.F.IF. EXAMPLE Writing Polnomials in Nested Form Write f() = in nested form. f () = Write the function. f () = ( ) - 6 Factor an out of the first three terms. f () = (( ) - 5) - 6 Factor an out of the first two terms in parentheses. REFLECT Houghton Mifflin Harcourt Publishing Compan a. Use the standard form and the nested form to evaluate the polnomial for = 3. Then compare both methods. Which is easier? Wh? Module 110 Lesson 1

3 3 MCC9 1.F.IF.7c EXAMPLE Graphing Polnomials Graph f() = A Determine the end behavior of the graph. The end behavior is determined b the leading term, 3. So, as +, f (), and as -, f (). B Complete the table of values. Use the nested form from the previous eample to evaluate the polnomial f() C Plot the points from the table on the graph, omitting an points whose -values are much greater than or much less than the other -values on the graph. D Draw a smooth curve through the plotted points, keeping in mind the end behavior of the graph REFLECT 3a. What are the zeros of the function? How can ou identif them from the graph? Houghton Mifflin Harcourt Publishing Compan 3b. What are the approimate values of for which the function is increasing? decreasing? 3c. A student wrote that f () has a minimum value of approimatel -8. Do ou agree or disagree? Wh? 3d. Without graphing, what do ou think the graph of g () = looks like? Wh? Module 111 Lesson 1

4 Now ou will sketch a variet of polnomial functions. You do not need to put values on the -ais. The emphasis is on showing the overall shape of the graph and its -intercepts. MCC9 1.F.IF.7c EXPLORE Investigating the Behavior of Graphs Near Zeros Use a graphing calculator to graph each function. Sketch the graphs on the aes provided below. Then complete the table. A f () = ( - 1)( - )( - 3)( - ) B g() = ( - 1) ( - )( - 3) C h() = ( - 1 ) 3 ( - ) D j() = ( - 1 ) E Eamining Zeros f() g() h() j() What are the zeros of the function? 1,, 3, How man times does each zero occur in the factorization? At which zero(s) does the graph cross the -ais? At which zero(s) is the graph tangent to the -ais? REFLECT 1: times, 3: 1 time 1, 1 Houghton Mifflin Harcourt Publishing Compan a. Based on our results, make a generalization about the number of times a zero occurs in the factorization of a function and whether the graph of the function crosses or is tangent to the -ais at that zero. Module 11 Lesson 1

5 The factored form of a polnomial is useful for graphing because the zeros can easil be determined. The degree of a polnomial in factored form is the sum of the degrees of the factors. 5 MCC9 1.F.IF.7c EXAMPLE Sketching the Graph of a Factored Polnomial Function Sketch the graph of f() = ( + ) ( + 1)( - )( - 3). A Determine the end behavior. The degree of the polnomial is the sum of the degrees of the factors. So, the degree of f () is. If ou multipl the factors to write f () standard form, a n n + a n - 1 n a 1 + a 0, the leading coefficient a n is. Because the degree is odd and the leading coefficient is positive, f () as + and f () as -. B Describe the behavior at the zeros. The zeros of the function are. Identif how man times each zero occurs in the factorization. Determine the zero(s) at which the graph crosses the -ais. Houghton Mifflin Harcourt Publishing Compan Determine the zero(s) at which the graph is tangent to the -ais. C Sketch the graph at right. Use the end behavior to determine where to start and end. You ma find it helpful to plot a few points between the zeros to help get the general shape of the graph. REFLECT a. Can ou determine how man times a zero occurs in the factorization of a polnomial function just b looking at the graph of the function? Eplain. Module 113 Lesson 1

6 prep for MCC9 1.A.APR.3 6 ENGAGE The Rational Zero Theorem If ou multipl the factors of the function f () = ( - 1)( - )( - 3)( - ), ou can write f () in standard form as follows. f () = So, f () is a polnomial with integer coefficients that begins with the term and ends with the term. The zeros of f () are 1,, 3, and. Notice that each zero is a factor of the constant term,. Now consider the function g() = ( - 1)(3 - )( - 3)(5 - ). If ou multipl the factors, ou can write g() in standard form as follows. g() = So, g() is a polnomial with integer coefficients that begins with the term 10 and ends with the term. The zeros of g() are 1,, 3, and. In this case, the numerator of each 3 5 zero is a factor of the constant term,, and the denominator of each zero is a factor of the leading coefficient, 10. These eamples illustrate the Rational Zero Theorem. Rational Zero Theorem If p() = a n n + a n - 1 n a + a 1 + a 0 has integer coefficients, then ever rational zero of p() is a number of the following form: c b = factor of constant term a 0 factor of leading coefficient a n REFLECT 7 6a. If c is a rational zero of a polnomial function p(), eplain wh b - c must be b a factor of the polnomial. MCC9 1.A.APR.3 EXAMPLE Using the Rational Zero Theorem Sketch the graph of f() = Houghton Mifflin Harcourt Publishing Compan A Use the Rational Zero Theorem to identif the possible rational zeros of f (). The constant term is 1. Integer factors of the constant term are ±1, ±, ±3, ±, ±6, and ±1. The leading coefficient is. Integer factors of the leading coefficient are. Module 11 Lesson 1

7 B the Rational Zero Theorem, the possible rational zeros of f () are all rational numbers of the form c where c is a factor of the constant term and b is a factor b of the leading coefficient. List all the possible rational zeros. Possible rational zeros: B Test the possible rational zeros until ou find one that is an actual zero. Use snthetic substitution to test 1 and So, is a zero, and therefore is a factor of f (). C Factor f () = completel. Use the results of the snthetic substitution to write f () as the product of a linear factor and a quadratic factor. f () = ( )( ) Factor the quadratic factor to write f () as a product of linear factors. f () = ( )( )( ) Use the factorization to identif the other zeros of f (). Houghton Mifflin Harcourt Publishing Compan How man times does each zero occur in the factorization? D Determine the end behavior. f () as + and f () as -. E Sketch the graph of the function on the coordinate plane at right. REFLECT 7a. How did ou determine where the graph crosses the -ais and where it is tangent to the -ais? Module 115 Lesson 1

8 7b. How did factoring the polnomial help ou graph the function? 7c. How did using the Rational Zero Theorem to find one zero help ou find the other zeros? practice Write each polnomial function in nested form. Then sketch the graph b plotting points and using end behavior. 1. f() = -. f() = f() = f() = f() = 6 f() = f() = f() = Houghton Mifflin Harcourt Publishing Compan Module 116 Lesson 1

9 5. Given the graph of a polnomial function, how can ou tell if a given zero occurs an even or an odd number of times? Sketch the graph of each factored polnomial function. 6. f () = ( - 3)( + ) 7. g() = ( + 1 ) h() = ( + 3)( + 1) ( - 1) 9. j() = ( + ) 3 ( - 3 ) Houghton Mifflin Harcourt Publishing Compan 10. From 000 to 010, the profit (in thousands of dollars) for a small business is modeled b P() = , where is the number of ears since 000. a. Sketch a graph of the function at right. b. What are the zeros of the function in the domain 0 10? c. What do the zeros represent? Module 117 Lesson 1

10 Use the Rational Zero Theorem to identif the possible zeros of each function. Then factor the polnomial completel. Finall, identif the actual zeros and sketch the graph of the function. 10. f () = Possible zeros: Factored form of function: Actual zeros: 11. g() = Possible zeros: Factored form of function: Actual zeros: 1. h() = Possible zeros: Factored form of function: Actual zeros: The polnomial function p() has degree 3, and its zeros are -3,, and 6. What do ou think is the equation of p()? Do ou think there could be more than one possibilit? Eplain. Houghton Mifflin Harcourt Publishing Compan Module 118 Lesson 1

11 Name Class Date -1 Additional Practice Solve each polnomial equation b factoring = = = = 0 Identif the roots of each equation. State the multiplicit of each root = = 0 Identif all the real roots of each equation = = 10 Solve. 9. An engineer is designing a storage compartment in a spacecraft. The compartment must be meters longer than it is wide and its depth must be 1 meter less than its width. The volume of the compartment must be 8 cubic meters. a. Write an equation to model the volume of the compartment. Houghton Mifflin Harcourt Publishing Compan b. List all possible rational roots. c. Use snthetic division to find the roots of the polnomial equation. Are the roots all rational numbers? d. What are the dimensions of the storage compartment? Module 119 Lesson 1

12 Problem Solving Most airlines have rules concerning the size of checked baggage. The rules for Budget Airline are such that the dimensions of the largest bag cannot eceed 5 in. b 55 in. b 6 in. A designer is drawing plans for a piece of luggage that athletes can use to carr their equipment. It will have a volume of 76,75 cubic inches. The length is 10 in. greater than the width and the depth is 1 in. less than the width. What are the dimensions of this piece of luggage? 1. Write an equation in factored form to model the volume of the piece of luggage.. Multipl and set the equation equal to zero. 3. Think about possible roots of the equation. Could a root be a multiple of? a multiple of 5? a multiple of 10?. How do ou know?. Use snthetic substitution to test possible roots. Choose positive integers that are factors of the constant term and reasonable in the contet of the problem. Possible Root ,75 Choose the letter for the best answer. 5. Which equation represents the factored polnomial? A (w + 55)(w + 5w ) = 0 B (w 35)(w + 60w + 105) = 0 C (w 5)(w + 1w ) = 0 D (w )(w 10w + 76,75) = 0 6. Which could be the dimensions of this piece of luggage? A 31 in. b 5 in. b 55 in. B 5 in. b 55 in. b 55 in. C 5 in. b 5 in. b 55 in. D 5 in. b 55 in. b 6 in. Houghton Mifflin Harcourt Publishing Compan Module 10 Lesson 1