8.2 Finding Complex Solutions of Polynomial Equations


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1 Locker LESSON 8. Finding Complex Solutions of Polynomial Equations Texas Math Standards The student is expected to: A.7.D Determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods. Also A.7.B, A.7.E Mathematical Processes A.1.F The student is expected to analyze mathematical relationships to connect and communicate mathematical ideas. Language Objective.D.1,.I.,.E,.H.,.G Complete a Solving Polynomial Equations chart with a partner. Name Class Date 8. Finding Complex Solutions of Polynomial Equations Essential Question: What do the Fundamental Theorem of Algebra and its corollary tell you about the roots of the polynomial equation p(x) = 0 where p(x) has degree n? A.7.D Determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods. Also A.7.B, A.7.E Explore Investigating the Number of Complex Zeros of a Polynomial Function You have used various algebraic and graphical methods to find the roots of a polynomial equation p (x) = 0 or the zeros of a polynomial function p (x). Because a polynomial can have a factor that repeats, a zero or a root can occur multiple times. The polynomial p (x) = x + 8 x + 1x + 18 = (x + ) (x + ) has  as a zero once and  as a zero twice, or with multiplicity. The multiplicity of a zero of p (x) or a root of p (x) = 0 is the number of times that the related factor occurs in the factorization. In this Explore, you will use algebraic methods to investigate the relationship between the degree of a polynomial function and the number of zeros that it has. Find all zeros of p (x) = x + 7x. Include any multiplicities greater than 1. p (x) = x + 7 x Factor out the GCF. p (x) = (x + 7) What are all the zeros of p(x)? 0 (mult. ), 7 x Resource Locker ENGAGE Essential Question: What do the Fundamental Theorem of Algebra and its corollary tell you about the roots of the polynomial equation p (x) = 0 where p (x) has degree n? The equation has exactly n complex roots provided that you count the multiplicities of the roots. Find all zeros of p (x) = x  6. Include any multiplicities greater than 1. p (x) = x  6 Factor the difference of two cubes. p (x) = (x  ) (x + x + 16 ) What are the real zeros of p (x)? Solve x + x + 16 = 0 using the quadratic formula. b ± b  ac a ± i 1 ±  ± PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and what variables you might use to describe the amount of violence in a movie. Then preview the Lesson Performance Task.  ± i What are the nonreal zeros of p (x)?  + i,  i Module 8 7 Lesson Name Class Date 8. Finding Complex Solutions of Polynomial Equations Essential Question: What do the Fundamental Theorem of Algebra and its corollary tell you about the roots of the polynomial equation p(x) = 0 where p(x) has degree n? A.7.D Determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods. Also A.7.B, A.7.E Explore Investigating the Number of Complex Zeros of a Polynomial Function You have used various algebraic and graphical methods to find the roots of a polynomial equation p (x) = 0 or the zeros of a polynomial function p (x). Because a polynomial can have a factor that repeats, a zero or a root can occur multiple times. The polynomial p (x) = x + 8 x + 1x + 18 = (x + ) (x + ) has  as a zero once and  as a zero twice, or with multiplicity. The multiplicity of a zero of p (x) or a root of p (x) = 0 is the number of times that the related factor occurs in the factorization. In this Explore, you will use algebraic methods to investigate the relationship between the degree of a polynomial function and the number of zeros that it has. Find all zeros of p (x) = x + 7x. Include any multiplicities greater than 1. p (x) = x + 7 x x 0 (mult. ), 7 Factor out the GCF. p (x) = (x + 7) What are all the zeros of p(x)? p (x) = x  6 Find all zeros of p (x) = x  6. Include any multiplicities greater than 1. Factor the difference of two cubes. p (x) = (x ) (x + + ) What are the real zeros of p (x)? Solve x + x + 16 = 0 using the quadratic formula. b ± b  ac a ± ±  ± i What are the nonreal zeros of p (x)?  x 16  ± _ i i,  i Resource Module 8 7 Lesson HARDCOVER PAGES 01 1 Turn to these pages to find this lesson in the hardcover student edition. 7 Lesson 8.
2 C Find all zeros of p (x) = x + x  x  1x. Include any multiplicities greater than 1. p (x) = x + x  x  1x Factor out the GCF. p (x) = x ( x ) + x  x  1 Group terms to begin p (x) = x ( ( x + x )  ( x + 1 )) factoring by grouping. Factor out common monomials. p (x) = x ( x (x + )  (x + ) ) Factor out the common binomial. p (x) = x (x + ) ( x  ) Factor the difference of squares. p (x) = x (x + ) ( x + )( x  ) What are all the zeros of p (x)? 0, , , D Find all zeros of p (x) = x Include any multiplicities greater than 1. p (x) = x  16 Factor the difference of squares. p (x) = ( x )  ( x + ) Factor the difference of squares. p (x) = ( x + )( x  )( x + ) What are the real zeros of p (x)? , EXPLORE Investigating the Number of Complex Zeros of a Polynomial Function INTEGRATE TECHNOLOGY Students have the option of completing the Explore activity either in the book or online. QUESTIONING STRATEGIES When would you need to use the quadratic formula to find a zero? When one of the factors of the polynomial is a nonfactorable quadratic polynomial. Solve x + = 0 by taking square roots. x + = 0 x =  ± _  ± i What are the nonreal zeros of p (x)? i, i Module 8 8 Lesson PROFESSIONAL DEVELOPMENT Learning Progressions Students have learned factoring techniques in earlier lessons, and a more general technique for finding zeros of polynomial functions and solutions of polynomial equations based on the Rational Zero/Root Theorem in the previous lesson. They have also learned how to use the quadratic formula to solve quadratic equations. In this lesson, students pull all these techniques together in order to understand and use the Fundamental Theorem of Algebra. Finding Complex Solutions of Polynomial Equations 8
3 INTEGRATE MATHEMATICAL PROCESSES Focus on Patterns Encourage students to look for patterns in their results. They can make connections between the degree of each polynomial and the number of zeros, and between a function s characteristics and their effects on the nature of its zeros. Students can also be prompted to make conjectures about the number of each type of zero (real and nonreal) that could exist for polynomials of varying degrees. E Find all zeros of p (x) = x + 5 x + 6 x x 8. Include multiplicities greater than 1. By the Rational Zero Theorem, possible rational zeros are ±1, ±, ±, and ±8. Use a synthetic division table to test possible zeros. m_ n The remainder is 0, so 1 is/is not a zero. p (x) factors as (x  1) ( ). Test for zeros in the cubic polynomial. m_ n x + 6 x + 1x a zero. p (x) factors as (x  1) (x + ) ( x ) + x +. The quadratic is a perfect square trinomial. So, p (x) factors completely as p (x) = (x  1) (x + ). 1,  (mult. ) What are all the zeros of p (x)? F Complete the table to summarize your results from Steps A E. Polynomial Function in Standard Form p (x) = x + 7 x p (x) = x  6 p (x) = x + x  x  1x p (x) = x  16 Polynomial Function Factored over the Integers p (x) = x (x + 7) p (x) = (x  ) ( x + x + 16) p (x) = x (x + ) (x + ) (x  ) p (x) = (x  ) (x + ) ( x + ) Real Zeros and Their Multiplicities 0 (mult. ) ; 7 0; ; ; ; Nonreal Zeros and Their Multiplicities None  + i ;   i None i; i p (x) = x + 5 x + 6 x  x  8 p (x) = (x  1) (x + ) 1,  (mult. ) None Module 8 9 Lesson COLLABORATIVE LEARNING PeertoPeer Activity Provide pairs of students with a fourth degree polynomial equation and a fifth degree polynomial equation. Have them work together to determine the number of possible combinations of types of roots for each equation. Then have them graph their equations, and use the graphs to help predict which combination of roots will be the correct combination for each function. Challenge them to solve the equations to verify their predictions. 9 Lesson 8.
4 Reflect 1. Examine the table. For each function, count the number of unique zeros, both real and nonreal. How does the number of unique zeros compare with the degree? The number of unique zeros is less than or equal to the degree.. Examine the table again. This time, count the total number of zeros for each function, where a zero of multiplicity m is counted as m zeros. How does the total number of zeros compare with the degree? The total number of zeros is the same as the degree of the function.. Discussion Describe the apparent relationship between the degree of a polynomial function and the number of zeros it has. The number of zeros of a polynomial function is the same as the degree of the function when you include complex zeros and count the multiplicities of the zeros in the total. Explain 1 Applying the Fundamental Theorem of Algebra to Solving Polynomial Equations The Fundamental Theorem of Algebra and its corollary summarize what you have observed earlier while finding rational zeros of polynomial functions and in completing the Explore. The Fundamental Theorem of Algebra Every polynomial function of degree n 1 has at least one zero, where a zero may be a complex number. EXPLAIN 1 Applying the Fundamental Theorem of Algebra to Solving Polynomial Equations INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections Substantiate The Fundamental Theorem of Algebra and its corollary by applying them to solutions of linear equations and easily factorable quadratic equations, with which students are familiar. Include examples of quadratic equations that have roots with multiplicity of. Corollary: Every polynomial function of degree n 1 has exactly n zeros, including multiplicities. Because the zeros of a polynomial function p (x) give the roots of the equation p (x) = 0, the theorem and its corollary also extend to finding all roots of a polynomial equation. Example 1 Solve the polynomial equation by finding all roots. x  1 x  x + 0 = 0 The polynomial has degree, so the equation has exactly roots. x  1 x  x + 0 = 0 Divide both sides by. x  6 x  17x + 10 = 0 Group terms. ( x  6 x )  (17x  10) = 0 Factor out common monomials. x (x  6)  17 (x  6) = 0 Factor out the common binomial. ( x  17) (x  6) = 0 One root is 6. Solving x  17 = 0 gives x = 17, or ± _ 17. The roots are  _ 17, _ 17, and 6. Module 8 0 Lesson DIFFERENTIATE INSTRUCTION Communicating Math Understanding the concept of the degree of a polynomial is important in applying the Fundamental Theorem of Algebra and its corollary. Students (especially English language learners) may benefit from a rigorous review of finding degrees of polynomials written in standard form, factored form, and with terms in varying orders of degree. Focus on polynomials that contain only singlevariable monomials. Check that students can explain how to find the degree of the polynomial for each of the different forms. Finding Complex Solutions of Polynomial Equations 0
5 QUESTIONING STRATEGIES If, after using synthetic substitution to test all possible rational roots of a cubic equation, you find only one root of the equation, can you conclude that the remaining roots are imaginary? Explain. No. The remaining roots may be imaginary or they may be irrational. B x  6 x  7 = 0 The polynomial has degree, so the equation has exactly roots. Notice that x  6 x  7 has the form u  6u  7, where u = x. So, you can factor it like a quadratic trinomial. x  6 x  7 = 0 Factor the trinomial. ( x  ) ( x + ) = 0 Factor the difference of squares. ( x + ) (x  ) ( x + ) = 0 The real roots are  and. Solving x + = 0 gives x = , or 9 ± _  = ± i. The roots are ,, i, i. Reflect. Restate the Fundamental Theorem of Algebra and its corollary in terms of the roots of equations. Theorem: For every polynomial of degree n 1, the equation p (x) = 0 has at least one root, where a root may be a complex number. Corollary: For every polynomial of degree n 1, the equation p (x) = 0 has exactly n roots, when you include multiplicity. Your Turn Solve the polynomial equation by finding all roots x  7 = 0 6. p (x) = x  1 x + 55 x  91x (x  ) ( x + 6x + 9) = 0 x  = 0 _ x + 6x + 9 = 0 (6) ± (6)  ()(9) () 6 ± 108 = 6 ± 6i ± i, or  _ ± _ i The roots are _ , + i, and   i. x ( x  1 x + 55x  91) = 0 One root is 0. Possible rational roots: ±1, ±7, ±1, ±91. Use synthetic division to test for roots. A second root is 7. Solve x  6x + 1 = 0. (6) ± (6)  (1)(1) 1 6 ± 16 = 6 ± i ± i The roots are 0, 7, + i, and  i. Module 8 1 Lesson LANGUAGE SUPPORT Communicate Math Have students work in pairs. Have them write the theorems in this module for solving polynomial equations, the Rational Zero Theorem, Rational Roots Theorem, and the Fundamental Theorem of Algebra, and then work together to explain the theorems in their own words. Then have students write the explanations and give an example for each theorem. 1 Lesson 8.
6 Explain Writing a Polynomial Function From Its Zeros You may have noticed in finding roots of quadratic and polynomial equations that any irrational or complex roots come in pairs. These pairs reflect the ± in the quadratic formula. For example, for any of the following number pairs, you will never have a polynomial equation for which only one number in the pair is a root. _ 5 and  _ 5 ; and 17 ; i and i; + 1i and  1i; _ _ 6 i and 11_ 61_ 6 i The irrational root pairs a + b c and a  b c are called irrational conjugates. The complex root pairs a + bi and a  bi are called complex conjugates. Irrational Root Theorem If a polynomial p (x) has rational coefficients and a + b c is a root of the equation p (x) = 0, where a and b are rational and c is irrational, then a  b c is also a root of p (x) = 0. Complex Conjugate Root Theorem If a + bi is an imaginary root of a polynomial equation with realnumber coefficients, then a  bi is also a root. Because the roots of the equation p (x) = 0 give the zeros of a polynomial function, corresponding theorems apply to the zeros of a polynomial function. You can use this fact to write a polynomial function from its zeros. Because irrational and complex conjugate pairs are a sum and difference of terms, the product of irrational conjugates is always a rational number and the product of complex conjugates is always a real number. (  _ 10 ) ( + _ 10 ) =  ( _ 10 ) =  10 = 6 (1  i _ ) (1 + i _ ) = 1  (i _ ) = 1  (1) () = EXPLAIN Writing a Polynomial Function From its Zeros QUESTIONING STRATEGIES If one zero of a fourth degree polynomial function is rational, what must be true about the other three zeros? One of the three must also be rational. The other two could be either irrational conjugates or imaginary conjugates. Is it possible for a fifth degree polynomial equation to have no real zeros? Explain. No. Since imaginary zeros occur in conjugate pairs, there could be at most imaginary zeros. Therefore, at least one zero must be real. Example Write the polynomial function with least degree and a leading coefficient of 1 that has the given zeros. 5 and + _ 7 Because irrational zeros come in conjugate pairs,  _ 7 must also be a zero of the function. Use the zeros to write a function in factored form, then multiply to write it in standard form. Multiply the first two factors using FOIL. p (x) = x  ( + _ 7 ) x  (  _ 7 ) (x  5) Multipy the conjugates. = x  (  _ 7 ) x  ( + _ 7 ) x + (97) (x  5) Combine like terms. = x + ( + _ _ 7 ) x + (19) (x  5) Simplify. = x  6x  19 (x  5) = x  (  _ 7 ) x  ( + _ 7 ) x + ( + _ 7 )(  _ 7 ) (x  5) Distributive property = x ( x  6x  19)  5 ( x  6x  19) AVOID COMMON ERRORS Students may make errors when multiplying factors of the form (x  a), where a is an irrational number such as + or an imaginary number such as 1  i. Encourage them to multiply each of these types of factors with the factor that contains the conjugate of the irrational or imaginary number first, and show them how to use grouping to make the multiplication easier. Multiply. = x  6 x  19x  5 x + 0x + 95 Combine like terms. = x  11 x + 11x + 95 The polynomial function is p (x) = x  11 x + 11x Module 8 Lesson Finding Complex Solutions of Polynomial Equations
7 INTEGRATE MATHEMATICAL PROCESSES Focus on Critical Thinking Have students discuss how they could write the rule for a third degree polynomial function whose graph passes through (1 +, 0) and the origin. Then have them find the function, and use a graphing calculator to check their work. B, and 1 i Because complex zeros come in conjugate pairs, must also be a zero of the function. Use the zeros to write a function in factored form, then multiply to write it in standard form. p (x) = x  (1 + i) x  ( 1  i ) (x  ) (x  ) Multiply the first = x  (1  i) x  two factors using FOIL. ( 1 + i ) x + (1 + i) (1  i) (x  ) (x  ) Multipy the conjugates. Combine like terms. = x + (1 + i i) x + (x  ) (x  ) Simplify. = ( ) (x  ) (x  ) Multipy the binomials. = ( x  x + ) Distributive property = x ( x  5x + 6)  x ( x  5x + 6) + ( x  5x + 6) Multipy. = ( x  5 x + 6 x ) + ( x + 10 x  1x) + ( x  10x + 1) Combine like terms. = = 1 + i x  (1  i) x  (1 + i) x + ( 1  ( x  x + ( x  5x + 6) x  7 x + 18 x  x The polynomial function is p (x) = x  7 x + 18 x  x + 1. )) (x  ) (x  ) Reflect 7. Restate the Irrational Root Theorem in terms of the zeros of polynomial functions. If a polynomial function p (x) has rational coefficients and a + b c is a zero of the function, where a and b are rational and c is irrational, then a  b c is also a zero of p (x). 8. Restate the Complex Conjugates Zero Theorem in terms of the roots of equations. If a + bi is an imaginary zero of a polynomial function p (x) with realnumber coefficients, then a  bi is also a zero of p (x). Module 8 Lesson Lesson 8.
8 Your Turn Write the polynomial function with the least degree and a leading coefficient of 1 that has the given zeros i and  7 _ The polynomial function must also have  i and + 7 as zeros. p (x) = x  ( + i) x  (  i) x  ( + 7 ) x  (  7 ) = x  (  i) x  ( + i) x + ( + i) (  i) x  (  7 x + ( + 7 ) (  7 ) = x  (  i) x  ( + i) x + (  9 (1)) x  (  7 ) x  ( + 7 ) x  ( + 7 x + (169 ) = x + ( + i   i) x + 1 x + ( ) x  8 = ( x  x + 1) ( x  8x  8) = x ( x  8x  8) x ( x  8x  8) + 1 ( x  8x  8) = ( x  8 x  8 x ) + ( x + x + 8x) + (1 x  10x ) = x  1 x  7 x + x The polynomial function is p (x) = x  1 x  7 x + x ) ) EXPLAIN Solving a RealWorld Problem by Graphing Polynomial Functions INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling Lead students to recognize that the solution of the problem is not a zero of either p (x) or q (x) ; however, it is a zero of the difference function p (x)  q (x). This can be confirmed from the graphs of the three functions. Explain Solving a RealWorld Problem by Graphing Polynomial Functions You can use graphing to help you locate or approximate any real zeros of a polynomial function. Though a graph will not help you find nonreal zeros, it can indicate that the function has nonreal zeros. For example, look at the graph of p (x) = x  x . The graph intersects the xaxis twice, which shows that the function has two real zeros. By the corollary to the Fundamental Theorem of Algebra, however, a fourth degree polynomial has four zeros. So, the other two zeros of p (x) must be nonreal. The zeros are  _, _, i, and i. (A polynomial whose graph has a turning point on the xaxis has a real zero of even multiplicity at that point. If the graph bends at the xaxis, there is a real zero of odd multiplicity greater than 1 at that point.)  y x Module 8 Lesson Finding Complex Solutions of Polynomial Equations
9 QUESTIONING STRATEGIES Why do the methods shown in Parts A and B produce the same solution? When you solve the equation p (x) = q (x), you are finding the value of x for which the two functions are equal. Since p(x) is equal to q (x) at this value of x, this is the value that would make their difference, p (x)  q (x), equal to 0. The following polynomial models approximate the total oil consumption C (in millions of barrels per day) for North America (NA) and the Asia Pacific region (AP) over the period from 001 to 011, where t is in years and t = 0 represents 001. C NA (t) = t t + 0. t  0.9t +.6 C AP (t) = t t + 1.t Use a graphing calculator to plot the functions and approximate the xcoordinate of the intersection in the region of interest. What does this represent in the context of this situation? Determine when oil consumption in the Asia Pacific region overtook oil consumption in North America using the requested method. Graph Y1 = x x + 0. x  0.9x +.6 and Y = x x + 1.x Use the Calc menu to find the point of intersection. Here are the results for Xmin = 0, Xma 10, Ymin = 0, Yma 0. (The graph for the Asia Pacific is the one that rises upward on all segments.) The functions intersect at about 5, which represents the year 006. This means that the models show oil consumption in the Asia Pacific equaling and then overtaking oil consumption in North America about 006. Find a single polynomial model for the situation in Example A whose zero represents the time that oil consumption for the Asia Pacific region overtakes consumption for North America. Plot the function on a graphing calculator and use it to find the xintercept. Let the function C D (t) represent the difference in oil consumption in the Asia Pacific and North America. the time that consumption is equal A difference of 0 indicates. C D (t) = C AP (t)  C NA (t) = t t + 1.t (0.009 t t + 0. t  0.9t +.6) Remove parentheses and rearrange terms. = t t t t  0. t + 1.t + 0.9t Combine like terms. Round to three significant digits. = t t t + 1.7t .50 Graph C D (t) and find the xintercept. (The graph with Ymin = , Yma 6 is shown.) Within the rounding error, the results for the xcoordinate of the intersection of C NA (t) and C AP (t) and the xintercept of C D (t) are the same. Module 8 5 Lesson 5 Lesson 8.
10 Your Turn 10. An engineering class is designing model rockets for a competition. The body of the rocket must be cylindrical with a coneshaped top. The cylinder part must be 60 cm tall, and the height of the cone must be twice the radius. The volume of the payload region must be 558π cm in order to hold the cargo. Use a graphing calculator to graph the rocket s payload volume as a function of the radius x. On the same screen, graph the constant function for the desired payload. Find the intersection to find x. Let V represent the volume of the payload region. V = V cone + V cylinder 1 V (x) = π x (x) + π x (60) = Elaborate π x + 60π x To find _ x when the volume is 558π, graph y = π x + 60π x and y = 558π and find the points of intersection. Because the radius must be positive, the radius of the rocket is cm. 11. What does the degree of a polynomial function p(x) tell you about the zeros of the function or the roots of the equation p (x) = 0? The degree tells you how many zeros or roots there are when you include complex zeros or roots and count the multiplicities of repeated zeros or roots. 1. A polynomial equation of degree 5 has the roots 0.,, 8, and 10.6 (each of multiplicity 1). What can you conclude about the remaining root? Explain your reasoning. The remaining root must be rational. This is because any irrational roots or imaginary roots always occur in conjugate pairs. So, if there were an irrational or imaginary root, there would have to be two of them. 1. Discussion Describe two ways you can use graphing to determine when two polynomial functions that model a realworld situation have the same value. You can graph both functions on the same coordinate grid and find the xvalue of any point where the two graphs intersect. Also, you can form a new function that is the difference of the two original functions. The xintercepts of the graph of this function will also be the xvalues where the original functions have the same value. 1. Essential Question CheckIn What are possible ways to find all the roots of a polynomial equation? By the corollary to the Fundamental Theorem of Algebra, you know that the number of roots equals the degree of the equation. You can factor when possible, and use the Rational Root theorem along with the Zero Product Property to find rational roots. You can use the quadratic formula to find irrational or complex roots. Module 8 6 Lesson ELABORATE INTEGRATE MATHEMATICAL PROCESSES Focus on Patterns Lead students to the generalization that a polynomial function of odd degree must have an odd number (counting repeated zeros) of real zeros and, in particular, must have at least one real zero. QUESTIONING STRATEGIES A fourth degree polynomial function has only the zeros ,, and. How can this be true given the requirement of the Corollary of the Fundamental Theorem of Algebra, which states that a polynomial of degree n has exactly n zeros? One of the zeros must occur twice. The corollary requires that repeated zeros be counted multiple times. INTEGRATE MATHEMATICAL PROCESSES Focus on Critical Thinking Ask students to discuss the possibility of two polynomial functions that model a realworld situation having more than one value for which they are equal. Have them discuss the implications of this situation on the graphs of the functions and on the graph of the difference function. SUMMARIZE THE LESSON How can you use the Fundamental Theorem of Algebra, its corollary, and the Irrational Conjugates and Complex Conjugates Theorems to determine the possible combinations of types of zeros of a polynomial function? You can use the Fundamental Theorem of Algebra and its corollary to find the total number of zeros of the function. Then you can use the fact that irrational and imaginary zeros occur in conjugate pairs to determine the possible combinations. Finding Complex Solutions of Polynomial Equations 6
11 EVALUATE Evaluate: Homework and Practice Find all zeros of p (x). Include any multiplicities greater than 1. Online Homework Hints and Help Extra Practice ASSIGNMENT GUIDE Concepts and Skills Explore Investigating the Number of Complex Zeros of a Polynomial Function Example 1 Applying the Fundamental Theorem of Algebra to Solving Polynomial Equations Example Writing a Polynomial Function From its Zeros Example Solving a RealWorld Problem by Graphing Polynomial Functions QUESTIONING STRATEGIES Practice Exercises 1 Exercises Exercises 5 8 Exercises 9 11 How does the Rational Zero Theorem help you find zeros that are not rational? The Rational Zero Theorem can be used to identify rational zeros and the corresponding factors. Then, other methods, such as the quadratic formula, may be used to find other zeros that are irrational or imaginary. 1. p (x) = x  10 x + 10x . p (x) = x  x + x  1 Possible rational zeros are ±1, ±, ±, ± 1_, ± _, ± _. is a zero. p (x) = (x  ) ( x  x + ) Solve x  x + = 0.  () ± ()  ()() () ± 8 ± i ± i = = 6 6 The zeros of p (x) are, + i, and  i. Solve the polynomial equation by finding all roots. p (x) = x  x + x  1 = ( x  x ) + (x  1) = x (x  ) + (x  ) = ( x + ) (x  ) is a zero. Solve x + = 0. x =  ±  = ±i The zeros of p (x) are, i, and i.. x  x + 8x  1 = 0. x  5 x + x + 0 ( x  x ) + (8x  1) = 0 x (x  ) + (x  ) = 0 (x + ) (x  ) = 0 _ x + = 0 x  = 0  ±  = ±i The roots are _, i, and i. x (x  5 x + x + 1) = 0 0 is a root. Possible rational roots are 1 and is a root. x (x  1) ( x  x  1) = 0 Solve x  x  1 = 0.  () ± ()  (1)(1) (1) ± 0 = ± 5 = ± 5 The roots are 0, 1, + 5, and  5. Module 8 7 Lesson Exercise Depth of Knowledge (D.O.K.) Mathematical Processes 1 8 Skills/Concepts 1.F Analyze relationships 9 10 Skills/Concepts 1.D Multiple representations 11 Skills/Concepts 1.A Everyday life 1 Strategic Thinking 1.F Analyze relationships 1 1 Strategic Thinking 1.F Analyze relationships 15 Strategic Thinking 1.F Analyze relationships 7 Lesson 8.
12 Write the polynomial function with least degree and a leading coefficient of 1 that has the given zeros. 5. 0, _ 5, and 6. i,, and  Because irrational zeros come in conjugate pairs,  5 must also be a zero. p (x) = x (x  5 ) (x + 5 )(x  ) = x ( x  5) (x  ) = x ( x  x  5x + 10) = x  x  5 x + 10x 7. 1, 1 (multiplicity ), and i 8. (multiplicity of ) and i Because complex zeros come in conjugate pairs, i must also be a zero. p (x) = (x  ) (x + ) (x  i) (x + i) = ( x  ) ( x + 16) = x + 1 x  6 Because complex zeros come in conjugate pairs, i must also be a zero. p (x) = (x  1) (x + 1) (x  i) (x + i) = (x  1) (x + 1) (x + 1) (x  i) (x + i) = ( x  1) ( x + x + 1) ( x + 9) = ( x + 8 x  9) ( x + x + 1) = x ( x + x + 1) + 8 x ( x + x + 1)  9 ( x + x +1) = x 6 + x 5 + x + 8 x + 16 x + 8 x  9 x  18x  9 = x 6 + x x + 16 x  x  18x  9 Because complex zeros come in conjugate pairs, i must also be a zero. p (x) = (x  ) (x  i) (x + i) = ( x  6x + 9) ( x + 9) = x ( x  6x + 9) + 9 ( x  6x + 9) = x  6 x + 9 x + 9 x  5x + 81 = x  6 x + 18 x  5x + 81 AVOID COMMON ERRORS Students often make sign errors when writing factors for zeros or roots that are irrational, such as  5, or imaginary, such as + i. Encourage them to use parentheses within parentheses when writing the factors, and to be careful to apply the distributive property when removing the parentheses or regrouping the terms. INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning Have students discuss why irrational roots of a polynomial equation with rational coefficients must occur in conjugate pairs. Have them consider the resulting polynomial if, for example, only one of three factors of a cubic polynomial equation contained an irrational number. Module 8 8 Lesson Finding Complex Solutions of Polynomial Equations 8
13 VISUAL CUES Have students graph several of the functions using a graphing calculator to provide a visual connection between each type of zero (rational, irrational, and imaginary), and its representation on the graph of the function. Help students to see how irrational zeros can be approximated from xintercepts. Lead them to observe that a function that has only imaginary zeros has no xintercepts. CRITICAL THINKING Students may be interested to find that they can test irrational and imaginary zeros of a polynomial function using synthetic substitution. Encourage them to use this process to check their work. 9. Forestry Height and trunk volume measurements from 10 giant sequoias between the heights of 0 and 75 feet in California give the following model, where h is the height in feet and V is the volume in cubic feet. V (h) = 0.11 h h + 1,00h  1,67,00 The President tree in the Giant Forest Grove in Sequoia National Park has a volume of about 5,100 cubic feet. Use a graphing calculator to plot the function V (h) and the constant function representing the volume of the President tree together. (Use a window of 0 to 75 for X and 0,000 to 55,000 for Y.) Find the xcoordinate of the intersection of the graphs. What does this represent in the context of this situation? 10. Business Two competing stores, store A and store B, opened the same year in the same neighborhood. The annual revenue R (in millions of dollars) for each store t years after opening can be approximated by the polynomial models shown. R A (t) = ( t + 1 t  77 t + 600t + 1,650) R B (t) = ( t + 6 t t + 68t + 90) The xcoordinate of the intersection gives the model s predicted height for a tree with the volume of the President tree. This predicted height is about 65 feet. Image Credits: RichardBakerUSA/Alamy Using a graphing calculator, graph the models from t = 0 to t = 10, with a range of 0 to for R. Find the xcoordinate of the intersection of the graphs, and interpret the graphs. Graph Y1 = ( x + 1 x  77 x + 600x + 1,650) for R A. Graph Y = ( x + 6 x x + 68x + 90) for R B. Then find the point of intersection. The functions intersect at 9, which corresponds to having the same annual revenue 9 years after the stores opened. Module 8 9 Lesson 9 Lesson 8.
14 11. Personal Finance A retirement account contains cash and stock in a company. The cash amount is added to each week by the same amount until week, then that same amount is withdrawn each week. The functions shown model the balance B (in thousands of dollars) over the course of the past year, with the time t in weeks. B C (t) = 0.1 t B S (t) = t t t t Use a graphing calculator to graph both models (Use 0 to 0 for range.). Find the xcoordinate of any points of intersection. Then interpret your results in the context of this situation. The graphs intersect at xvalues of about 8 and 7. This means that at those weeks of the year, the cash balance and stock balance in the account were the same. LANGUAGE SUPPORT Connect Vocabulary Remind students that they learned complex numbers have a real and an imaginary part. The complex conjugate of a + bi is a  bi, and similarly the complex conjugate of a  bi is a + bi. This consists of changing the sign of the imaginary part of a complex number. The real part is left unchanged. 1. Match the roots with their equation. A. 1 A, B, E, F x + x + x + x  8 = 0 B.  A, B, C, D x  5 x + = 0 C. D. 1 E. i F. i x + x + x + x  8 = 0 in factored form is (x  1) (x + ) ( x + ) = 0. Roots are 1, , i, and i. x  5 x + = 0 in factored form is (x + 1) (x  1) (x + ) (x  ) = 0. Roots are 1, 1, , and. Module 8 0 Lesson Finding Complex Solutions of Polynomial Equations 0
15 PEERTOPEER DISCUSSION Ask students to discuss with a partner why, although the Rational Root Theorem can always be used to help find the roots of a cubic equation, it may not be useful for finding the roots of a fourth degree polynomial equation. Since a cubic equation has three roots, at least one of them will be rational (since irrational and imaginary roots occur in conjugate pairs). The other two roots, no matter what type, can be found by factoring or by using the quadratic formula. A fourth degree equation will have four roots, none of which may be rational, so the Rational Root Theorem may not be of help. JOURNAL Have students describe how they would go about finding the roots of a fifth degree polynomial equation if they know that at least two of the roots are rational. H.O.T. Focus on Higher Order Thinking 1. Draw Conclusions Find all of the roots of x 6 5 x 15x + 15,65 = 0. (Hint: Rearrange the terms with a sum of cubes followed by the two other terms.) ( x ,65)  5 x  65 x = 0 ( x ) x  65 x = 0 ( x + 5) ( x  5 x + 65)  5 x ( x + 5) = 0 ( x + 5) ( x  5 x x ) = 0 ( x + 5) ( x 50 x + 65) = 0 ( x + 5) ( x  5) = 0 ( x + 5) (x+ 5) (x  5) = 0 The roots are 5 and 5, each with multiplicity, and 5i and 5i. 1. Explain the Error A student is asked to write the polynomial function with least degree and a leading coefficient of 1 that has the zeros 1 + i, 1  i, _, and . The student writes the product of factors shown, and multiplies them together to obtain p (x) = x + (1  _ ) x  ( + _ )x + (6 + _ ) x  6 _. What error did the student make? What is the correct function? The function must have 5 zeros. The zero must be paired with its conjugate, . p (x) = x  (1 + i) x  (1  i) (x  ) (x + )(x + ) = x  (1  i) x (1 + i) x + (1 + i) (1  i) ( x  ) (x + ) = x + (1 + i 1  i) x + (1 (1)) ( x + x x  6) = ( x  x + ) ( x + x  x  6) = ( x 5 + x  x  6 x ) + ( x  6 x + x + 1x) + ( x + 6 x  x  1) = x 5 + x  6x + x + 8x Critical Thinking What is the least degree of a polynomial equation that has i as a root with a multiplicity of, and  _ as a root with multiplicity? Explain. The least degree is 10. Since i is a root times, then i must also be a root times. Since  is a root times, then + must also be a root times, and = 10. Module 8 1 Lesson 1 Lesson 8.
16 Lesson Performance Task In 198 the MPAA introduced the PG1 rating to their movie rating system. Recently, scientists measured the incidences of a specific type of violence depicted in movies. The researchers used specially trained coders to identify the specific type of violence in one half of the top grossing movies for each year since The trend in the average rate per hour of 5minute segments of this type of violence in movies rated G/PG, PG1, and R can be modeled as a function of time by the following equations: V G/PG (t) = t V PG1 (t) = t t + 0.6t V R (t) =.15 V is the average rate per hour of 5minute segments containing the specific type of violence in movies, and t is the number of years since a. Interestingly, in 1985 or t = 0, V G/PG (0) > V PG1 (0). Can you think of any reasons why this would be true? b. What do the equations indicate about the relationship between V G/PG (t) and V PG1 (t) as t increases? c. Graph the models for V G/PG (t) and V PG1 (t) and find the year in which V PG1 (t) will be greater than V G/PG (t). a. Possible answers include but are not limited to The rating of PG1 was poorly understood by the people responsible for rating the films. Films released in the years immediately following 1985 had been scripted, filmed, and/or edited before the rating was fully understood by the film studios, so they hadn t separated the specific type of violence out of the G/PG movies. b. The equations indicate that as t increases, V PG1 (t) will eventually be greater than V G/PG (t). V G/PG (t) is a linear function with a negative first term so its end behavior on the right is decreasing to negative infinity while the leading term of V PG1 (t) is positive, so its end behavior on the right is increasing to infinity. c. The functions intersect at a value of t, which indicates that the average rate per hour of 5minute segments of violence in movies rated PG1 first surpassed the average hourly rate in movies rated G/PG in Rate per Hour 1 V(t) V R V G/PG Years Since 1985 V PG 1 t Connect Vocabulary Students may not be familiar with the abbreviations of the movie rating system. Explain that the abbreviations indicate how appropriate the movie is for difference audiences. A G rating means the movie is for General audiences. A PG rating means Parental Guidance is suggested. A PG1 rating means Parental Guidance is suggested and the movie may not be appropriate for children under age 1. An R rating means entrance is Restricted; an adult must accompany children under 17. AVOID COMMON ERRORS Students may think that the models V (t) give the total amount of violence in a movie. Ask students what the units of V (t) are. number of 5minute segments per hour Ask students how to calculate the total minutes of violence in a movie. Multiply V (t) by 5 and then multiply by the length of the movie in hours. INTEGRATE MATHEMATICAL PROCESSES Focus on Communication Discuss with students why V PG1 increases to infinity as t increases. Ask them if it makes sense that V PG1 becomes greater than V R and whether they think this will actually happen. Have students explain how they could create a model that would more accurately predict V PG1 for future years.. Module 8 Lesson EXTENSION ACTIVITY Have students research the topgrossing movie for each year since 1985 and whether it was rated G, PG, PG1, or R. Have students discuss whether the success of a movie is related to its rating. Ask them if they think the amount of violence in a movie makes it more or less popu lar. Scoring Rubric points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Finding Complex Solutions of Polynomial Equations
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