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LESSON 7.2 Name Finding Complex Solutions of Polynomial Equations Class 7.2 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? Common Core Math Standards A-APR.2 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. 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). Also N-CN.9(+), A-REI.11, A-APR.3, F-IF.7c The polynomial p(x) = x 3 + 8x 2 + 21x + 18 = (x + 2)(x + 3) has -2 as a zero once and -3 as a zero twice, or with multiplicity 2. 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. 2 Mathematical Practices 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. MP.7 Using Structure Language Objective A Find all zeros of p(x) = x 3 + 7x 2. Include any multiplicities greater than 1. p(x) = x 3 + 7x 2 Complete a “Solving Polynomial Equations” chart with a partner. Factor out the GCF. 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. p(x) = x 2 (x + 7) ) ( What are all the zeros of p(x)? 0 mult. 2 , -7 © Houghton Mifflin Harcourt Publishing Company ENGAGE The equation has exactly n complex roots provided that you count the multiplicities of the roots. B Find all zeros of p(x) = x 3 - 64. Include any multiplicities greater than 1. p(x) = x 3 - 64 p(x) = x - 4 x 2 + 4x + 16 What are the real zeros of p(x)? 4 ) Solve x + 4x + 16 = 0 using the quadratic formula. 2 ――― -b ± √b - 4ac x = __ 2a 2 ――― ―――――― √ √ _ -4 ± 4 - 4 ∙ 1 ∙ 16 -4 ± -48 -4 ± 4i √3 x = ___ x = _x = __ 2 2 2∙ 1 2 ― x = -2 ± 2i √3 ― ― √ √ What are the non-real zeros of p(x)? -2 + 2i 3 , -2 -2i 3 Module 7 Lesson 2 353 gh “File info” made throu be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction Date Class s lution mplex So Finding Co ial Equations of Polynom Name 7.2 Resource Locker tell you corollary ra and its p(x) has em of Algeb 0 where mental Theor equation p(x) = do the Funda polynomial ion: What N-CN.9(+), roots of the CA2. Also about the g on page table startin degree n? rd, see the of this standa the full text A-APR.2 For of Complex .3, F-IF.7c Number A-REI.11, A-APR Quest Essential A2_MNLESE385894_U3M07L2.indd 353 HARDCOVER PAGES 255264 g the n InvestigatinPolynomial Functio mial a of a polyno a the roots Zeros of can have Explore Turn to these pages to find this lesson in the hardcover student edition. mial ds to find se a polyno cal metho n p(x). Becau and graphi s algebraic mial functio used variou of a polyno multiple times. You have once or the zeros 2 p(x) = 0 root can occur -2 as a zero equation zero or a + 3) has a root of repeats, a x + 2)(x of p(x) or + 18 = ( factor that of a zero 3 x2 + 21x multiplicity the factorization. =x +8 in licity 2. The mial p(x) factor occurs or with multip n the The polyno a zero twice, times that the related nship betwee and -3 as gate the relatio number of investi the is to ds = 0 has. p(x) that it aic metho r of zeros use algebr e, you will and the numbe r than 1. function In this Explor mial greate licities a polyno degree of e any multip 2 Find all zeros What are y g Compan of p(x) = the GCF. Factor out Find all zeros Publishin Factor the Harcour t What are of p(x)? difference the real zeros 2 4x + 16 Solve x + n Mifflin 0 (mult. 2), -7 r than 1. licities greate e any multip x3 - 64. Includ 3 - 64 p(x) = x 16 2 + 4x + - 4 x p(x) = x of two cubes. of p(x) = atic formu the quadr 2 © Houghto x= ± -b _ 2a L2.indd ―――――― √ What are 4_U3M07 ) la. ――― _ √3 -4 ± 4i __ -4 ± -48 x = 2 ∙ 16 2 4 -4∙ 1 =_ 2 -4 ± __ x _ 2∙ 1 Module 7 SE38589 )( ( 4 of p(x)? = 0 using ― ―― - 4ac √b_ x = -2 ± A2_MNLE Includ x3 + 7x . 2 3 + 7x p(x) = x 2 (x + 7) p(x) = x all the zeros x= Lesson 7.2 )( ( Factor the difference of two cubes. 353 Resource Locker Investigating the Number of Complex Zeros of a Polynomial Function Explore The student is expected to: 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? Date 353 ― 2i √3 eal zeros the non-r of p(x)? √ -2 + 2i ― √3 , -2 -2i ― √3 Lesson 2 353 3/19/14 3:24 PM 3/19/14 3:23 PM C Find all zeros of p(x) = x 4 + 3x 3 - 4x 2 - 12x. Include any multiplicities greater than 1. p(x) = x 4 + 3x 3 - 4x 2 - 12x Factor out the GCF. Group terms to begin factoring by grouping. ( p(x) = x x 3 + 3x 2 - 4x - 12 ( ( p(x) = x (x 3 + 3x 2) - 4x + 12 ( )) Factor out common monomials. p(x) = x x 2 (x + 3) - 4 (x + 3) Factor out the common binomial. p(x) = x(x + 3)(x 2 - 4) Factor the difference of squares. p(x) = x(x + 3) What are all the zeros of p(x)? 0, -3, -2, 2 D ) ( x+2 )( EXPLORE Investigating the Number of Complex Zeros of a Polynomial Function ) x-2 INTEGRATE TECHNOLOGY Students have the option of completing the Explore activity either in the book or online. ) QUESTIONING STRATEGIES Find all zeros of p(x) = x 4 - 16. Include any multiplicities greater than 1. p(x) = x 4 - 16 Factor the difference of squares. Factor the difference of squares. ( p x =( p(x) = ( ) x2 - 4 x+2 What are the real zeros of p(x)? -2, 2 When would you need to use the quadratic formula to find a zero? When one of the factors of the polynomial is a non-factorable quadratic polynomial. ) (x + 4) )( x - 2 )(x + 4) 2 2 Solve x 2 + 4 = 0 by taking square roots. x2 + 4 = 0 x 2 = -4 _ x = ±√ -4 © Houghton Mifflin Harcourt Publishing Company x = ± 2i What are the non-real zeros of p(x)? -2i, 2i Module 7 354 Lesson 2 PROFESSIONAL DEVELOPMENT Learning Progressions A2_MNLESE385894_U3M07L2.indd 354 3/19/14 3:23 PM 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 354 E INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Encourage students to look for patterns in Find all zeros of p(x) = x 4 + 5x 3 + 6x 2 -4x -8. Include multiplicities greater than 1. By the Rational Zero Theorem, possible rational zeros are ±1, ±2, ±4, and ±8. Use a synthetic division table to test possible zeros. 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 non-real) that could exist for polynomials of varying degrees. m _ n 1 5 6 -4 -8 1 1 6 12 8 0 The remainder is 0, so 1 is/is not a zero. ( ) p(x) factors as (x - 1) x 3 + 6x 2 + 12x + 8 . Test for zeros in the cubic polynomial. m _ n 1 6 12 8 1 1 7 19 27 -1 1 5 7 1 2 1 8 28 64 -2 1 4 4 0 -2 a zero. ( ) p(x) factors as (x - 1)(x + 2) x 2 + 4x + 4 . The quadratic is a perfect square trinomial. (x + 2)3 . 1, -2 (mult. 3) What are all the zeros of p(x)? So, p(x) factors completely as p(x) = (x - 1) © Houghton Mifflin Harcourt Publishing Company F Complete the table to summarize your results from Steps A–E. Polynomial Function in Standard Form Real Zeros and Their Multiplicities Polynomial Function Factored over the Integers Non-real Zeros and Their Multiplicities p(x) = x 3 + 7x 2 p(x) = x 2 (x + 7) 0 (mult. 2); -7 p(x) = x 3 - 64 p(x) = (x - 4)(x 2 + 4x + 16) 4 p(x) = x 4 + 3x 3 - 4x 2 - 12x p(x) = x(x + 3)(x + 2)(x - 2) 0; -3; -2; 2 None p(x) = x - 16 p(x) = (x - 2)(x + 2)(x + 4) -2; 2 -2i; 2i p(x) = x 4 + 5x 3 + 6x 2 - 4x - 8 p(x) = (x - 1)(x + 2) 1, -2 (mult. 3) None 4 2 Module 7 3 355 None ― -2 - 2i √― 3 -2 + 2i √3 ; Lesson 2 COLLABORATIVE LEARNING A2_MNLESE385894_U3M07L2.indd 355 Peer-to-Peer 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. 355 Lesson 7.2 3/19/14 3:23 PM Reflect EXPLAIN 1 1. Examine the table. For each function, count the number of unique zeros, both real and non-real. How does the number of unique zeros compare with the degree? The number of unique zeros is less than or equal to the degree. 2. 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. 3. 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 Applying the Fundamental Theorem of Algebra to Solving Polynomial Equations INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Substantiate The Fundamental Theorem of 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 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 2. 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. 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 © Houghton Mifflin Harcourt Publishing Company Solve the polynomial equation by finding all roots. 2x 3 - 12x 2 - 34x + 204 = 0 The polynomial has degree 3, so the equation has exactly 3 roots. 2x 3 - 12x 2 - 34x + 204 = 0 x 3 - 6x 2 - 17x + 102 = 0 Divide both sides by 2. (x 3 - 6x 2) - (17x - 102) = 0 Group terms. Factor out common monomials. x 2(x - 6) - 17(x - 6) = 0 (x 2 - 17)(x - 6) = 0 Factor out the common binomial. _ One root is x = 6. Solving x 2 - 17 = 0 gives x 2 = 17, or x = ±√17 . _ _ The roots are -√17 , √17 , and 6. Module 7 356 Lesson 2 DIFFERENTIATE INSTRUCTION A2_MNLESE385894_U3M07L2.indd 356 3/19/14 3:23 PM 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 single-variable 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 356 B QUESTIONING STRATEGIES x 4 - 6x 2 - 27 = 0 The polynomial has degree 4 , so the equation has exactly 4 roots. 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. Notice that x 4 - 6x 2 - 27 has the form u 2 - 6u - 27, where u = x 2. So, you can factor it like a quadratic trinomial. x 4 - 6x 2 - 27 = 0 (x - 9 )(x + 3 ) = 0 2 Factor the trinomial. (x + 3 )(x - 3 )(x + 3) = 0 2 Factor the difference of squares. The real roots are -3 and 2 3 ―― _ x = ±√-3 = ± i 3 . 3 , i √― 3 -3, 3, -i √― The roots are . Solving x 2 + 3 = 0 gives x 2 = -3 , or . Reflect 4. 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 © Houghton Mifflin Harcourt Publishing Company Solve the polynomial equation by finding all roots. 5. 8x 3 - 27 = 0 6. (2x - 3)(4x 2 + 6x + 9) = 0 x(x 3 - 13x 2 + 55x - 91) = 0 2x - 3 = 0 One root is x = 0. 3 x=_ Possible rational roots: ±1, ±7, ±13, ±91. 2 4x 2 + 6x + 9 = 0 -(6) ± ――――― Use synthetic division to test for roots. √(6)2 - 4(4)(9) x = ____________________ A second root is x = 7. 2(4) -6 ± √―― -108 -6 ± 6i √― 3 x = ___________ = ________ 8 8 ― -3 ± 3i √3 3 √― 3 x = ________ , or -_ ±_ i 3 4 4 4 ― √ -3 + 3i 3 -3 - 3i √― 3 3 _ ________ ________ The roots are 2, p(x) = x 4 - 13x 3 + 55x 2 - 91x 4 , and 4 Solve x 2 - 6x + 13 = 0. ―― 2 -(-6) ± (-6) - 4(1)(13) x = _______________________ 2∙1 ―― 6 ± 4i = _____ x = _________ 2 2 6 ± √-16 . x = 3 ± 2i The roots are 0, 7, 3 + 2i, and 3 - 2i. Module 7 357 Lesson 2 LANGUAGE SUPPORT A2_MNLESE385894_U3M07L2 357 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. 357 Lesson 7.2 6/27/14 10:53 PM Explain 2 Writing a Polynomial Function From Its Zeros EXPLAIN 2 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 ― ― ― ― Writing a Polynomial Function From its Zeros _ 1 i √3 and _ 1 i √3 11 + _ 11 - _ and -√ 5 ; 1 + √7 and 1 - √7 ; i and -i; 2 + 14i and 2 - 14i; _ 6 6 6 6 ― ― 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 QUESTIONING STRATEGIES ― 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. ― ― 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. Complex Conjugate Root Theorem If a + bi is an imaginary root of a polynomial equation with real-number coefficients, then a - bi is also a root. 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 4 imaginary zeros. Therefore, at least one zero must be real. 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. _ _ _ (2 - √10 )(2 + √10 ) = 22 - (√10 )2 = 4 - 10 = -6 _ _ _ (1 - i√2 )(1 + i√2 ) = 12 - (i√2 )2 = 1 - (-1)(2) = 3 Example 2 Write the polynomial function with least degree and a leading coefficient of 1 that has the given zeros. AVOID COMMON ERRORS _ 5 and 3 + 2√7 Multipy the conjugates. Combine like terms. Simplify. _ _ p(x) = ⎡⎣x - (3 + 2√7 )⎤⎦ ⎡⎣x - (3 - 2√7 )⎤⎦ (x - 5) = ⎡⎣x 2 - (3 - 2√7 )x - ( 3 + 2√7 ) x + ( 3 + 2√7 )( 3 - 2√7 )⎤⎦(x - 5) _ _ _ _ = ⎡⎣x 2 - (3 - 2√7 )x - (3 + 2√7 )x + (9 - 4 ⋅ 7)⎤⎦(x - 5) _ _ = ⎡⎣x 2 + (-3 + 2√7 - 3 - 2√7 )x + (-19)⎤⎦(x - 5) _ _ = ⎡⎣x 2- 6x - 19⎤⎦(x - 5) Distributive property = x(x 2 - 6x - 19) - 5(x 2 - 6x - 19) Multiply. = x 3 - 6x 2 - 19x - 5x 2 + 30x + 95 Combine like terms. = x 3 - 11x 2 + 11x + 95 ― © Houghton Mifflin Harcourt Publishing Company Multiply the first two factors using FOIL. Students may make errors when multiplying factors of the form (x - a), where a is an irrational number such as 3 + √2 or an imaginary number such as 1 - 4i. 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. _ Because irrational zeros come in conjugate pairs, 3 - 2√7 must also be a zero of the function. Use the 3 zeros to write a function in factored form, then multiply to write it in standard form. The polynomial function is p(x) = x 3 - 11x 2 + 11x + 95. Module 7 A2_MNLESE385894_U3M07L2 358 358 Lesson 2 6/27/14 10:54 PM Finding Complex Solutions of Polynomial Equations 358 B INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Have students discuss how they could write 2, 3 and 1- i Because complex zeros come in conjugate pairs, 1 + i must also be a zero of the function. Use the 4 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 - 2)(x - 3) ⎣ ― the rule for a third degree polynomial function whose graph passes through (1 + √2 , 0) and the origin. Then have them find the function, and use a graphing calculator to check their work. Multiply the first two factors using FOIL. Multipy the conjugates. ( ) ⎤ ⎡ = ⎢x 2 - (1 - i)x - 1 + i x + (1 + i)(1 - i)⎥ (x - 2)(x - 3) ⎣ ⎦ ( ( )) ⎡ = ⎢x 2 - (1 - i)x - (1 + i)x + 1 - -1 ⎣ ⎤ ⎥(x - 2)(x - 3) ⎦ Combine like terms. 2 = ⎡⎣x + (-1 + i - 1 - i)x + 2⎤⎦(x - 2)(x - 3) Simplify. = x 2 - 2x + 2 (x - 2)(x - 3) Multipy the binomials. = (x 2 - 2x + 2) Distributive property ( ) (x 2 - 5x + 6) = x2(x2 - 5x + 6) - 2x (x2 - 5x + 6) + 2(x2 - 5x + 6) Multipy. = (x4 - 5x3 + 6x2) + (-2x 3 + 10x2 - 12x) + (2x 2 - 10x + 12) Combine like terms. = x 4 - 7x 3 + 18x 2 - 22x + 12 4 3 2 The polynomial function is p(x) = x - 7x + 18x - 22x + 12 . 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 ― ― © Houghton Mifflin Harcourt Publishing Company 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 real-number coefficients, then a - bi is also a zero of p(x). Module 7 A2_MNLESE385894_U3M07L2 359 359 Lesson 7.2 359 Lesson 2 6/27/14 10:54 PM Your Turn EXPLAIN 3 Write the polynomial function with the least degree and a leading coefficient of 1 that has the given zeros. 9. _ 2 + 3i and 4 - 7√ 2 ― Solving a Real-World Problem by Graphing Polynomial Functions The polynomial function must also have 2 - 3i and 4 + 7 √2 as zeros. p(x) = ⎡⎣x - (2 + 3i)⎤⎦⎡⎣x - (2 - 3i)⎤⎦⎡⎣x - (4 + 7√2 )⎤⎦⎡⎣x - (4 - 7√2 )⎤⎦ = ⎡⎣x 2 - (2 - 3i)x - (2 + 3i)x + (2 + 3i)(2 - 3i)⎤⎦⎡⎣x 2 - (4 - 7√2 )x - (4 + 7√2 ) x + (4 + 7√2 )(4 - 7√2 )⎤⎦ = ⎡⎣x 2 - (2 - 3i)x - (2 + 3i)x + (4 - 9(-1))⎤⎦⎡⎣x 2 - (4 - 7√2 )x - (4 + 7√2 ) x + (16 - 49 ∙ 2)⎤⎦ = ⎡⎣x 2 + (-2 + 3i - 2 - 3i)x + 13⎤⎦⎡⎣x 2 + (-4 + 7√2 - 4 - 7√2 )x - 82⎤⎦ ― ― ― ― ― ― ― ― = (x 2 - 4x + 13)(x 2 - 8x - 82) INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 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. = x 2(x 2 - 8x - 82) -4x(x 2 - 8x - 82) + 13(x 2 - 8x - 82) = (x 4 - 8x 3 - 82x 2) + (-4x 3 + 32x 2 + 328x) + (13x 2 - 104x - 1066) = x 4 - 12x 3 - 37x 2 + 224x - 1066 The polynomial function is p(x) = x 4 - 12x 3 - 37x 2 + 224x - 1066. Explain 3 Solving a Real-World 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 non-real zeros, it can indicate that the function has non-real zeros. For example, look at the graph of p(x) = x 4 - 2x 2 - 3. Module 7 A2_MNLESE385894_U3M07L2.indd 360 360 y 2 x -4 -2 0 -2 2 4 © Houghton Mifflin Harcourt Publishing Company The graph intersects the x-axis 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 non-real. The zeros are -√3 , √3 , i, and -i. (A polynomial whose graph has a turning point on the x-axis has a real zero of even multiplicity at that point. If the graph “bends” at the x-axis, there is a real zero of odd multiplicity greater than 1 at that point.) 4 Lesson 2 3/19/14 3:22 PM Finding Complex Solutions of Polynomial Equations 360 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 2001 to 2011, where t is in years and t = 0 represents 2001. C NA(t) = 0.00494t 4 - 0.0915t 3 + 0.442t 2 - 0.239t + 23.6 C AP(t) = 0.00877t 3 - 0.139t 2 + 1.23t + 21.1 Use a graphing calculator to plot the functions and approximate the x-coordinate 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 = 0.00494x 4 - 0.0915x 3 + 0.442x 2 - 0.239x + 23.6 and Y2 = 0.00877x 3 - 0.139x 2 + 1.23x + 21.1. Use the “Calc” menu to find the point of intersection. Here are the results for Xmin = 0, Xmax = 10, Ymin = 20, Ymax = 30. (The graph for the Asia Pacific is the one that rises upward on all segments.) The functions intersect at about x = 5, which represents the year 2006. This means that the models show oil consumption in the Asia Pacific equaling and then overtaking oil consumption in North America about 2006. Find a single polynomial model for the situation in Example 3A 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 x-intercept. Let the function C D(t) represent the difference in oil consumption in the Asia Pacific and North America. A difference of 0 indicates the time that consumption is equal . C D(t) = C AP(t) - C NA(t) = 0.00877t - 0.139t 2 + 1.23t + 21.1 - (0.00494t 4 - 0.0915t 3 + 0.442t 2 - 0.239t + 23.6) 3 © Houghton Mifflin Harcourt Publishing Company Remove parentheses and rearrange terms. = -0.00494t 4 + 0.00877t 3 + 0.0915t 3 - 0.139t 2 - 0.442t 2 + 1.23t + 0.239t + 21.1 - 23.6 Combine like terms. Round to three significant digits. = -0.00494t 4 + 0.100t 3 - 0.581t 2 + 1.47t - 2.50 Graph C D(t) and find the x-intercept. (The graph with Ymin = -4, Ymax = 6 is shown.) Within the rounding error, the results for the x-coordinate of the intersection of C NA(t) and C AP(t) and the x-intercept of C D(t) are the same. Module 7 A2_MNLESE385894_U3M07L2 361 361 Lesson 7.2 361 Lesson 2 16/10/14 11:07 AM Your Turn ELABORATE 10. An engineering class is designing model rockets for a competition. The body of the rocket must be cylindrical with a cone-shaped 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 3 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. INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Lead students to the generalization that a Let V represent the volume of the payload region. V = Vcone + Vcylinder V(x) = _1 πx (2x) + πx (60) = _2 πx 2 2 3 3 3 To find x when the volume is 558π, graph 2 y = πx 3 + 60πx 2 and y = 558π and 3 find the points of intersection. 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. + 60πx 2 _ QUESTIONING STRATEGIES Because the radius must be positive, the radius of the rocket is 3 cm. A fourth degree polynomial function has only the zeros -2, 3, and 4. 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. Elaborate 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. 12. A polynomial equation of degree 5 has the roots 0.3, 2, 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, INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Ask students to discuss the possibility of two there would have to be two of them. point where the two graphs intersect. Also, you can form a new function that is the difference of the two original functions. The x-intercepts of the graph of this function will also be the x-values where the original functions have the same value. 14. Essential Question Check-In 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 © Houghton Mifflin Harcourt Publishing Company 13. Discussion Describe two ways you can use graphing to determine when two polynomial functions that model a real-world situation have the same value. You can graph both functions on the same coordinate grid and find the x-value of any polynomial functions that model a real-world 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. 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 SUMMARIZE THE LESSON use the quadratic formula to find irrational or complex roots. Module 7 A2_MNLESE385894_U3M07L2.indd 362 362 Lesson 2 6/28/14 12:37 AM 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 362 EVALUATE Evaluate: Homework and Practice • Online Homework • Hints and Help • Extra Practice Find all zeros of p(x). Include any multiplicities greater than 1. 1. p(x) = 3x 3 - 10x 2 + 10x - 4 2. 1 Possible rational zeros are ±1, ±2, ±4, ±_ , 3 2 4 _ _ ± ,± . 3 ASSIGNMENT GUIDE 3 Concepts and Skills Practice Explore Investigating the Number of Complex Zeros of a Polynomial Function Exercises 1–2 Example 1 Applying the Fundamental Theorem of Algebra to Solving Polynomial Equations Exercises 3–4 Example 2 Writing a Polynomial Function From its Zeros Exercises 5–8 Example 3 Solving a Real-World Problem by Graphing Polynomial Functions Exercises 9–11 = x 2(x - 3) + 4(x - 3) = (x 2 + 4)(x - 3) x= ―――――― 3 is a zero. (-4) ± √(-4) - 4(3)(2) ___________________ 2 ―― 2(3) ― ― = _______ = ______ x = _______ 6 6 3 4 ± √-8 4 ± 2i √2 2 ± i √2 2 + i √― 2 ______ 3 , and Solve x 2 + 4 = 0. x 2 = -4 The zeros of p(x) are 3, -2i, and 2i. . 3 ―― x = ±√-4 = ±2i 2 - i √― 2 ______ Solve the polynomial equation by finding all roots. 3. 2x 3 - 3x 2 + 8x - 12 = 0 4. x 2 (2x - 3) + 4(2x - 3) = 0 2 0 is a root. + 4)(2x - 3) = 0 Possible rational roots are 1 and -1. 2x - 3 = 0 3 x=_ 1 is a root. x(x - 1)(x 2 - 4x - 1) = 0 2 x2 + 4 = 0 © Houghton Mifflin Harcourt Publishing Company x 2 = -4 Solve x 2 - 4x - 1 = 0. -(-4) ± ―― Exercise 2 2 ― ― The roots are 0, 1, 2 + √5 , and 2 - √5 . Lesson 2 363 Depth of Knowledge (D.O.K.) 2(1) 4 ± √― 20 4 ± 2 √― 5 5 x = ______ = ______ = 2 ± √― 3 The roots are _ , -2i, and 2i. 2 Module 7 ―――――― √(-4)2 - 4(1)(-1) x = ____________________ x = ±√-4 = ±2i A2_MNLESE385894_U3M07L2.indd 363 x 4 - 5x 3 + 3x 2 + x = 0 x(x 3 - 5x 2 + 3x + 1) = 0 (2x 3 - 3x 2) + (8x - 12) = 0 (x 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. Lesson 7.2 = (x 3 - 3x 2) + (4x - 12) Solve 3x 2 - 4x + 2 = 0. The zeros of p(x) are 2, QUESTIONING STRATEGIES 363 p(x) = x 3 - 3x 2 + 4x - 12 p(x) = (x - 2)(3x 2 - 4x + 2) 2 is a zero. p(x) = x 3 - 3x 2 + 4x - 12 Mathematical Practices 1–8 2 Skills/Concepts MP.2 Reasoning 9–10 2 Skills/Concepts MP.6 Precision 11 2 Skills/Concepts MP.4 Modeling 12 3 Strategic Thinking MP.2 Reasoning 13–14 3 Strategic Thinking MP.2 Reasoning 15 3 Strategic Thinking MP.2 Reasoning 3/19/14 3:22 PM Write the polynomial function with least degree and a leading coefficient of 1 that has the given zeros. 5. 0, _ √5 , and 2 6. ― Because irrational zeros come in conjugate pairs, -√5 must also be a zero. ― ― p(x) = x(x - √5 )(x + √5 )(x - 2) = x(x - 5)(x - 2) 2 7. 3 Because complex zeros come in conjugate pairs, -4i must also be a zero. p(x) = (x - 2)(x + 2)(x - 4i)(x + 4i) = x 4 + 12x 2 - 64 2 = x - 2x - 5x + 10x 4 ― Students often make sign errors when writing factors for zeros or roots that are irrational, such as 2 - √5 , or imaginary, such as 2 + 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. 4i, 2, and -2 = (x 2 - 4)(x 2 + 16) = x(x - 2x - 5x + 10) 3 AVOID COMMON ERRORS 2 1, -1 (multiplicity 3), and 3i Because complex zeros come in conjugate pairs, -3i must also be a zero. 3 p(x) = (x - 1)(x + 1) (x - 3i)(x + 3i) INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Have students discuss why irrational roots of 2 = ⎡⎣(x - 1)(x + 1)⎤⎦ (x + 1) (x - 3i)(x + 3i) = (x 2 - 1)(x 2 + 2x + 1)(x 2 + 9) = (x 4 + 8x 2 - 9)(x 2 + 2x + 1) = x 4 (x 2 + 2x + 1) + 8x 2 (x 2 + 2x + 1) - 9(x 2 + 2x +1) 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. = x 6 + 2x 5 + x 4 + 8x 4 + 16x 3 + 8x 2 - 9x 2 - 18x - 9 = x 6 + 2x 5 + 9x 4 + 16x 3 - x 2 - 18x - 9 8. 3(multiplicity of 2) and 3i Because complex zeros come in conjugate pairs, -3i must also be a zero. 2 p(x) = (x - 3) (x - 3i)(x + 3i) = (x 2 - 6x + 9)(x 2 + 9) © Houghton Mifflin Harcourt Publishing Company = x 2(x 2 - 6x + 9) + 9(x 2 - 6x + 9) = x 4 - 6x 3 + 9x 2 + 9x 2 - 54x + 81 = x 4 - 6x 3 + 18x 2 - 54x + 81 Module 7 A2_MNLESE385894_U3M07L2.indd 364 364 Lesson 2 3/19/14 3:22 PM Finding Complex Solutions of Polynomial Equations 364 9. 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 x-intercepts. Lead them to observe that a function that has only imaginary zeros has no x-intercepts. Forestry Height and trunk volume measurements from 10 giant sequoias between the heights of 220 and 275 feet in California give the following model, where h is the height in feet and V is the volume in cubic feet. V(h) = 0.131h 3 - 90.9h 2 + 21,200h - 1,627,400 The “President” tree in the Giant Forest Grove in Sequoia National Park has a volume of about 45,100 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 220 to 275 for X and 30,000 to 55,000 for Y.) Find the x-coordinate of the intersection of the graphs. What does this represent in the context of this situation? The x-coordinate of the intersection gives the model’s predicted height for a tree with the volume of the President tree. This predicted height is about 265 feet. 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. 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) = 0.0001(-t 4 + 12t 3 - 77t 2 + 600t + 13,650) © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©RichardBakerUSA/Alamy R B(t) = 0.0001(-t 4 + 36t 3 - 509t 2 + 3684t + 3390) Using a graphing calculator, graph the models from t = 0 to t = 10, with a range of 0 to 2 for R. Find the x-coordinate of the intersection of the graphs, and interpret the graphs. Graph Y1 = 0.0001(-x 4 + 12x 3 - 77x 2 + 600x + 13,650) for R A. Graph Y2 = 0.0001(-x 4 + 36x 3 - 509x 2 + 3684x + 3390) for R B. Then find the point of intersection. The functions intersect at x = 9, which corresponds to having the same annual revenue 9 years after the stores opened. Module 7 A2_MNLESE385894_U3M07L2 365 365 Lesson 7.2 365 Lesson 2 6/27/14 11:03 PM 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 32, 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. 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. B C(t) = -0.12|t - 32| + 13 B S(t) = 0.00005t 4 - 0.00485t 3 + 0.1395t 2 - 1.135t + 15.75 Use a graphing calculator to graph both models (Use 0 to 20 for range.). Find the x-coordinate of any points of intersection. Then interpret your results in the context of this situation. The graphs intersect at x-values of about 38 and 47. This means that at those weeks of the year, the cash balance and stock balance in the account were the same. 12. Match the roots with their equation. A, B, E, F A. 1 B. -2 x 4 + x 3 + 2x 2 + 4x - 8 = 0 A, B, C, D x 4 - 5x 2 + 4 = 0 C. 2 D. -1 E. 2i F. -2i x 4 + x 3 + 2x 2 + 4x - 8 = 0 in factored form is (x - 1)(x + 2)(x 2 + 4) = 0. Roots are 1, -2, 2i, and -2i. Module 7 A2_MNLESE385894_U3M07L2.indd 366 366 © Houghton Mifflin Harcourt Publishing Company x 4 - 5x 2 + 4 = 0 in factored form is (x + 1)(x - 1)(x + 2)(x - 2) = 0. Roots are -1, 1, -2, and 2. Lesson 2 3/19/14 3:21 PM Finding Complex Solutions of Polynomial Equations 366 PEER-TO-PEER DISCUSSION H.O.T. Focus on Higher Order Thinking 13. Draw Conclusions Find all of the roots of x 6− 5x − 125x 2 + 15,625 = 0. (Hint: Rearrange the terms with a sum of cubes followed by the two other terms.) 4 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 will not be of help. (x 6 + 15,625)- 25x 4 - 625x 2 = 0 ⎡(x 2) 3+ 25 3⎤ - 25x 4 - 625x 2 = 0 ⎣ ⎦ (x 2 + 25)(x 4 - 25x 2 + 625) - 25x 2 (x 2 + 25)= 0 (x 2 + 25)(x 4 - 25x 2 + 625 - 25x 2) = 0 (x 2 + 25)(x 4 -50x 2 + 625) = 0 (x 2 + 25)(x 2 - 25) 2 = 0 2 (x 2 + 25)⎡⎣(x+ 5)(x - 5)⎤⎦ = 0 The roots are -5 and 5, each with multiplicity 2, and -5i and 5i. 14. 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, √2 , and -3. The student writes the product of factors them together to obtain _ _ shown, and multiplies _ _ p(x) = x 4 + (1 - √2 )x 3 - (4 + √2 )x 2 + (6 + 4√2 )x - 6√2 . What error did the student make? What is the correct function? ― ― The function must have 5 zeros. The zero √2 must be paired with its conjugate, - √2 . p(x) = ⎡⎣x - (1 + i)⎤⎦⎡⎣x - (1 - i)⎤⎦(x - √2 )(x + √2 )(x + 3) = ⎡⎣x 2 - (1 - i)x -(1 + i)x + (1 + i)(1 - i)⎤⎦(x 2 - 2)(x + 3) = ⎡⎣x 2 + (-1 + i -1 - i) x + (1 -(-1))⎤⎦(x 3 + 3x 2 -2x - 6) = (x 2 - 2x + 2)(x 3 + 3x 2 - 2x - 6) 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. ― = (x 5 + 3x 4 - 2x 3 - 6x 2) + (-2x 4 - 6x 3 + 4x 2 + 12x) + (2x 3 + 6x 2 - 4x - 12) © Houghton Mifflin Harcourt Publishing Company = x 5 + x 4 - 6x 3 + 4x 2 + 8x - 12 of a polynomial equation that has 3i as a 15. Critical Thinking What is the least degree _ root with a multiplicity of 3, and 2 - √3 as a root with multiplicity 2? Explain. ― Module 7 A2_MNLESE385894_U3M07L2.indd 367 367 Lesson 7.2 ― The least degree is 10. Since 3i is a root 3 times, then -3i must also be a root 3 times. Since 2 - √3 is a root 2 times, then 2 + √3 must also be a root 2 times, and 3 + 3 + 2 + 2 = 10. 367 Lesson 2 3/19/14 3:21 PM Lesson Performance Task 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 PG-13 rating means Parental Guidance is suggested and the movie may not be appropriate for children under age 13. An R rating means entrance is Restricted; an adult must accompany children under 17. In 1984 the MPAA introduced the PG-13 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 1985. The trend in the average rate per hour of 5-minute segments of this type of violence in movies rated G/PG, PG-13, and R can be modeled as a function of time by the following equations: V G/PG(t) = -0.015t + 1.45 V PG-13(t) = 0.000577t 3 - 0.0225t 2 + 0.26t + 0.8 V R(t) = 2.15 V is the average rate per hour of 5-minute segments containing the specific type of violence in movies, and t is the number of years since 1985. b. What do the equations indicate about the relationship between V G/PG(t) and V PG-13(t) as t increases? c. Graph the models for V G/PG(t) and V PG-13(t) and find the year in which V PG-13(t) will be greater than V G/PG(t). Rate per Hour a. Interestingly, in 1985 or t = 0, V G/PG(0) > V PG-13 (0). Can you think of any reasons why this would be true? 3 2 V(t) VR VPG–13 AVOID COMMON ERRORS VG/PG 1 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 5-minute 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 minutes. t 0 6 12 18 24 Years Since 1985 a. Possible answers include but are not limited to •The rating of PG-13 was poorly understood by the people responsible for rating the films. © Houghton Mifflin Harcourt Publishing Company •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 PG-13(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 PG-13(t) is positive, so its end behavior on the right is increasing to infinity. c. The functions intersect at a value of t ≈ 3, which indicates that the average rate per hour of 5-minute segments of violence in movies rated PG-13 first surpassed the average hourly rate in movies rated G/PG in 1988. Module 7 368 INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Discuss with students why V PG-13 increases to infinity as t increases. Ask them if it makes sense that V PG-13 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 PG-13 for future years.. Lesson 2 EXTENSION ACTIVITY A2_MNLESE385894_U3M07L2.indd 368 Have students research the top-grossing movie for each year since 1985 and whether it was rated G, PG, PG-13, 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 popular. 6/28/14 12:44 AM Scoring Rubric 2 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 368