7.1 Finding Rational Solutions of Polynomial Equations

LESSON 7.1 Finding Rational Solutions of Polynomial Equations Name 7.1 A-APR.B.2 Mathematical Practices COMMON CORE Finding Rational Solutions of Polynomial Equations Explore The student is expected to: 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 A-APR.B.3, A-CED.A.3 Relating Zeros and Coefficients of Polynomial Functions Identify the zeros of the polynomial function. B Multiply the factors to write the function in standard form. f ( x) = ( x + 2) (x - 1)(x + 3 ) = ( x2 + 2x - x - 2) (x + 3 ) = ( x2 + x - 2 ) (x + 3 ) Explain to a partner how to identify the factors of a polynomial function. = x 3 + 3 x 2 + x 2 + 3x - 2x - 6 Use the Rational Root Theorem to identify possible rational roots. Check each by using synthetic  = x 3 + 4 x2 + x - 6 How are the zeros of ƒ( x) related to the standard form of the function? Each of the zeros of the polynomial function is a factor of the constant term in the standard form.  Now consider the polynomial function g( x) = ( 2x + 3)( 4x - 5) (6x - 1) . Identify the zeros of this function. 3 The zeros are x = -_ , and x = 1_ . , x = 5_ 2 4 6  Multiply the factors to write the function in standard form. g( x) = ( 2x + 3) (4x - 5) (6x - 1) = (8x 2 - 10x + 12x - 15)( 6x - 1) substitution. If a rational root is found, repeat the process on the quotient obtained from the bottom row of the synthetic substitution. Continue to find rational roots in this way until the quotient is quadratic, at which point you can try factoring to identify the last two rational roots. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how the number of tourists in any given year can vary depending on many factors. Then preview the Lesson Performance Task. The zeros are x = -2, x = 1, and x = -3.  Language Objective Essential Question: How do you find the rational roots of a polynomial equation? = (8x 2 + 2x - 15)( 6x - 1 ) = 48x 3 - 8x2 + 12x 2 - 2x - 90x + 15  = 48x 3 + 4x2 - 92x + 15 How are the zeros of g( x) related to the standard form of the function? Each of the numerators of the zeros is a factor of the constant term, 15, and each of the denominators is a factor of the leading coefficient, 48. Module 7 Lesson 1 Date tions onal Solu Finding Rati ial Equations of Polynom Class Name 7.1 al equation? polynomi roots of a a, the l(p x) and a number. Also the rational you find For a polynomia p x( ) : How do Theorem: a fac tor of Question (x - a) is Remainder Essential apply the and only if Know and p , so p( a) = 0 if COMMON A-APR.B.2 by x - a is ( a) CORE on division remainder A-APR.B.3, cients of and Coeffi ng Zeros the Relatiomial Functions Consider are related. Polyn the function A-CED.A.3 Explore and the coefficient l function + 3.) of a polynomia ( x + 2)(x -1 () x The zeros ƒx ) l function ( = polynomia Identify the  B RDCOVER PAGES 247254 Resource Locker Multiply polynomi zeros of the the factors ( s of The zeros are x = -2, x = 1, and Turn to these pages to ind this lesson in the hardcover student edition. x = -3. al function. function to write the ) )( x + 3 in standard form. ) x-1 ) +2 (f x =( x ( 3) 2) x + 2 +x2 - x + ) 2 x - 2() x 3 the = (x + 2 zeros of x -2x -6 2 Each of the + x +3 = x 3 +3 x function? form of the 2 +x -6 form. the standard =3 x+4 x standard x related to term in the of ƒ( ) the zeros of the constant the zeros is a factor () 6x - 1.)Identify ial function ( 4x - 5 = 2( x + 3) x(g ) al function the polynomi Now consider _, and x = 16_. 5 of this function. _ 3 2 , x =4 are x = The zeros form. in standard function write the to 1 ) the factors Multiply ( - 5 ) 6x ( ) 4x x -1 ) x =( 2x + 3 - 15)( 6 g () 2 - x +12x = 8 (x 10 )1 ( x2 2x- 15) 6 15 (x + =8 90x + 2 - x 2 3 2x +12x = 48x -8 function? of the + 15 form of the and each 3 2 -92x term, 15, the standard = 48x +4x related to the constant ofx( g) factor of the zeros zeros is a How are ors of the nt, 48. numerat coefficie Each of the the leading factor of ators is a denomin = (x  How are polynom    341 Module 7 341 Lesson 7.1 Resource Locker The zeros of a polynomial function and the coefficients of the function are related. Consider the polynomial function ƒ( x) = (x + 2 )( x -1 )( x + 3) . MP.2 Reasoning ENGAGE Date Essential Question: How do you find the rational roots of a polynomial equation? Common Core Math Standards COMMON CORE Class Lesson 1 Reflect 1. In general, how are the zeros of a polynomial function related to the function written in standard form? Each of the numerators of the zeros is a factor of the constant term. Each of the 2. Discussion Does the relationship from the first Reflect question hold if the zeros are all integers? Explain. Yes; If the zeros are all integers, each of them can be written with a denominator of 1. Each 3. 1 3 5 __ x - __ If you use the zeros, you can write the factored form of g ( x) as g( x) = x + __ 2 x- 4 6 , rather than as g( x )= ( 2x + 3)( 4x - 5) (6x - 1) . What is the relationship of the factors between the two forms? Give this relationship in a general form. In each factor, the denominator of the fraction becomes the coefficient of the variable. EXPLORE Relating Zeros and Coefficients of Polynomial Functions denominators of the zeros is a factor of the leading coefficient. of the numerators is still a factor of the constant term. ( )( )( INTEGRATE TECHNOLOGY ) Students have the option of completing the Explore activity either in the book or online. In general, if the zero is - b_ a, the factor can be written as (ax + b ). Explain 1 QUESTIONING STRATEGIES Finding Zeros Using the Rational Zero Theorem What is the relationship between the factors of a polynomial function and the zeros of the function? The zeros are the values of x found by setting each factor equal to 0 and solving for x. If a polynomial function p( x) is equal to ( a 1x + b 1)( a 2x + b 2)( a 3x + b 3) , where a1, a 2 , a3, b1 , b 2, and b3 are integers, the leading coefficient of p (x )will be the product a 1a2 a 3 and the constant term will be b b b 3 the product b 1b 2b 3. The zeros of p (x) will be the rational numbers -a__11 , -a__22 , -__ a 3. Comparing the zeros of p (x ) to its coefficient and constant term shows that the numerators of the polynomial’s zeros are factors of the constant term and the denominators of the zeros are factors of the leading coefficient. This result can be generalized as the Rational Zero Theorem. 7 , what If a zero of a polynomial function is ___ 13 do you know about the coefficients when the polynomial is written in standard form? 7 is a factor of the constant term and 13 is a factor of the leading coefficient. Rational Zero Theorem ( ) m is a zero of p( x) p( m __) = 0 , If p( x ) is a polynomial function with integer coefficients, and if _ n n then m is a factor of the constant term of p ( x) and n is a factor of the leading coefficient of p(x). Example 1  Find the rational zeros of the polynomial function; then write the function as a product of factors. Make sure to test the possible zeros to find the actual zeros of the function. EXPLAIN 1 ƒ( x) = x 3 + 2x2 - 19x - 20 a. Use the Rational Zero Theorem to find all possible rational zeros. Factors of -20: ±1, ±2, ±4, ±5, ±10, ±20 Module 7 _ m 1 2 -19 -20 1 1 3 -16 -36 2 1 4 -11 -42 4 1 6 5 0 5 1 7 16 60 n 342 i b. Test the possible zeros. Use a synthetic division table to organize the work. In this table, the first row (shaded) represents the coefficients of the polynomial, the first column represents the divisors, and the last column represents the remainders. Lesson 1 PROFESSIONAL DEVELOPMENT Finding Zeros Using the Rational Zero Theorem QUESTIONING STRATEGIES Is every zero of a polynomial function represented in the set of numbers given by the Rational Zero Theorem? No. The Rational Zero Theorem gives only those zeros that are rational numbers. A polynomial function can also have zeros that are irrational numbers or imaginary numbers. Integrate Mathematical Practices This lesson provides an opportunity to address Mathematical Practice MP.2, which calls for students to translate between multiple representations and to “reason abstractly and quantitatively.” Students explore the relationship between the factors of a polynomial function and its zeros. They learn how to identify the zeros given the factors, and the factors given the zeros. They then explore the relationships between the rational zeros of a function and its leading coefficient and constant term, establishing the Rational Zero Theorem. Finding Rational Solutions of Polynomial Equations 342 c. Factor the polynomial. The synthetic division by 4 results in a remainder of 0, so 4 is a zero and the polynomial in factored form is given as follows: AVOID COMMON ERRORS Some students may forget to include 1 and –1 in their list of possible rational zeros. You may want to suggest that they write these first so that they are not inadvertently left off the list. ( x - 4) (x 2 + 6x + 5) = 0 (x - 4 )( x + 5)( x + 1) = 0 x = 4, x = -5, or x = -1 The zeros are x = 4, x = -5, and x = -1. B ƒ( x ) = x 4 - 4x3 - 7x 2 + 22x + 24 a. Use the Rational Zero Theorem to find all possible rational zeros. QUESTIONING STRATEGIES If the leading coefficient of a polynomial function with integer coefficients is 1, what can you conclude about the function’s rational zeros? Explain your reasoning. They must be integers, because when you apply the Rational Zero m Theorem, n can equal only 1 or –1 in ___ n. Factors of 24: ± 1 ,± 2 ,± 3 ,± 4 ,± 6 ,± 8 , ± 12 , ± 24 b. Test the possible zeros. Use a synthetic division table. _m 1 -4 -7 22 24 1 1 -3 -10 12 36 2 1 -2 -11 0 24 3 1 -1 -10 -8 0 n c. Factor the polynomial. The synthetic division by 3 results in a remainder of 0, so 3 is a zero and the polynomial in factored form is given as follows: (x - 3 10 x - ) (x 3 - x2 - 8 )=0 d. Use the Rational Zero Theorem again to find all possible rational zeros of g (x ) = x 3 - x 2 Factors of -8: ± 10 x - 8 . 1 ,± 2 ,± 4 ,± 8 e. Test the possible zeros. Use a synthetic division table. _m 1 -1 -10 -8 1 1 0 -10 -18 2 1 1 -8 -24 4 1 3 2 0 n f. Factor the polynomial. The synthetic division by 4 results in a remainder of 0, so 4 is a zero and the polynomial in factored form is: (x - 3 )(x - 4 (x - 3 )(x - 4 x= 3 ,x= 4 )( 1 )( x + x2 + 2 3 )(x + x+ 1 2 )= 0 )=0 , x = -2 , or x = -1 The zeros are x = 3, x = 4, x = -2, and x = -1. Module 7 343 Lesson 1 COLLABORATIVE LEARNING Small Group Activity Have students work in groups of 3–4 students. Instruct each group to create a fifth-degree polynomial function with rational zeros, not all of which are integers. Ask them to write their functions in standard from. Have groups exchange functions, and have each group create a poster showing how to apply the Rational Zero Theorem to find the zeros of the function. Students’ posters should also show verification that each number is indeed a zero of the function. 343 Lesson 7.1 Reflect 4. How is using synthetic division on a 4th degree polynomial to find its zeros different than using synthetic division on a 3 rd degree polynomial to find its zeros? To find the zeros of a 4 th degree polynomial using synthetic division, you need to use synthetic division to reduce that polynomial to a 3 rd degree polynomial and then use synthetic division again to reduce that polynomial to a quadratic polynomial that can be factored, if possible. 5. Suppose you are trying to find the zeros the function ƒ( x) = x 2 + 1. Would it be possible to use synthetic division on this polynomial? Why or why not? It would not be possible to find the zeros of this polynomial using synthetic substitution because the function has no rational roots, only complex roots. 6. 1 3 2 Using synthetic division, you find that __ 2 is a zero of ƒ ( x) = 2x + x - 13x + 6. The quotient () from the synthetic division array for ƒ __12 is 2x2 + 2x - 12. Show how to write the factored form of ƒ( x ) = 2x 3 + x 2 - 13x + 6 using integer coefficients. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Remind students that a zero of a function is a number from the domain that the function pairs with 0. Discuss that, for this reason, a graph of the function will have an x-intercept at each zero. Students can then make a concrete connection between the rational zeros they identify for a function, and the role the zeros play in the graph of the function. Using 1__ as a zero and the quotient 2x 2 + 2x - 12 you can write f( x) = 2x 3 + x2 - 13x + 6 2 1 ( 2x2 + 2x - 12 ) . as f( x) = x - _ 2 ) ) ( ( _) 1 f( x) = x - _ ( 2x2 + 2x - 12 ) = x - 12 ( 2)( x2 + x - 6) 2 ( = (2x - 1)(x2 + x - 6) = (2x - 1) (x + 3) (x - 2 ) Your Turn 7. Find the zeros of ƒ( x ) = x 3 + 3x2 - 13x- 15. a. Use the Rational Zero Theorem. Factors of -15: ±1, ±3, ±5, ±15 b. Test the possible zeros to find one that is actually a zero. m n _ 1 -3 -13 -15 1 1 4 -9 -24 3 1 6 5 0 c. Factor the polynomial using 3 as a zero. (x - 3 )( x2 + 6x + 5) = 0 ( x - 3) (x + 1) (x + 5) = 0 x = 3, x = -1, or x = -5 Module 7 The zeros are x = 3, x = -1, and x = -5. 344 Lesson 1 DIFFERENTIATE INSTRUCTION Visual Cues Encourage students to circle the leading coefficient in the function and to write “n is a factor of ” above it, and to circle the constant term in the function and to write “m is a factor of” above it. This will be helpful when applying the Rational Zero Theorem, and will keep students from erroneously writing the reciprocals of the possible rational zeros, especially since the usages of m and n appear in reverse alphabetical order with respect to the function. Finding Rational Solutions of Polynomial Equations 344 EXPLAIN 2 Solving a Real-World Problem Using the Rational Root Theorem Explain 2 Solving a Real-World Problem Using the Rational Root Theorem Since a zero of a function ƒ(x ) is a value of x for which ƒ( x ) = 0, finding the zeros of a polynomial function p( x) is the same thing as find the solutions of the polynomial equation p( x) = 0. Because a solution of a polynomial equation is known as a root, the Rational Zero Theorem can be also expressed as the Rational Root Theorem. Rational Root Theorem CONNECT VOCABULARY Explain how the words zeros and roots (or solutions) have similar meanings but are used in different contexts. The zeros of a function are the roots (or solutions) of the related equation. QUESTIONING STRATEGIES Why is it necessary to rewrite the equation so that it is equal to 0? In order to find the roots of an equation using the Rational Root Theorem, the equation must be in the form p(x ) = 0. What information is obtained by applying the Rational Zero Theorem to a polynomial function? A list of all possible rational zeros of the function If the polynomial p(x ) has integer coefficients, then every rational root of __ the polynomial equation p (x ) = 0 can be written in the form m n , where m is a factor of the constant term of p( x) and n is a factor of the leading coefficient of p (x ). Engineering A pen company is designing a gift container for their new premium pen. The marketing department has designed a pyramidal box with a rectangular base. The base width is 1 inch shorter than its base length and the height is 3 inches taller than 3 times the base length. The volume of the box must be 6 cubic inches. What are the dimensions of the box? Graph the volume function and the line y = 6 on a graphing calculator to check your solution. in tory g His arkin m the A. Analyze Information 1 The important information is that the base width must be inch shorter than 3 the base length, the height must be inches taller than 3 times the base length, and the box must have a volume of 6 cubic inches. B. Formulate a Plan Write an equation to model the volume of the box. Let x represent the base length in inches. The base width is 3x + 3 3( x + 1) height is , or . 1 _ ℓw h = V 3 x 1 () 1 6 1 ) 3 (x + ( ) (x = ) 3 1 x3- 1 x - 6 = 0 x-1 and the _ Module 7 345 Lesson 7.1 345 Lesson 1 C. Solve INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Prompt students to recognize that any Use the Rational Root Theorem to find all possible rational roots. Factors of -6: ± 1 , ± 2 , ± 3 , ± 6 Test the possible roots. Use a synthetic division table. _ m 1 0 -1 -6 1 1 1 0 -6 2 1 2 3 0 3 1 3 8 18 n Factor the polynomial. The synthetic division by so ( 2 1 2 rational roots found by factoring the resulting quadratic polynomial must be numbers that were identified as possible rational roots initially. This may help them to catch errors in factoring, or in performing the synthetic division. results in a remainder of 0, is a root and the polynomial in factored form is as follows: 3 ) =0 x - 2 ) ( 1 x2 + 2 x + complex The quadratic polynomial produces only roots, so the only possible 2 1 answer for the base length is inches. The base width is inch and the 9 inches. height is D. Justify and Evaluate The x-coordinates of the points where the graphs of two functions, f and g, intersect is the solution of the equation f(x ) = g( x). Using a graphing calculator to graph the volume function and y = 6 results in the graphs intersecting at the point (2, 6) . Since the x-coordinate is 2 , the answer is correct. Your Turn 8. Engineering A box company is designing a new rectangular gift container. The marketing department has designed a box with a width 2 inches shorter than its length and a height 3 inches taller than its length. The volume of the box must be 56 cubic inches. What are the dimensions of the box? A. The box width must be 2 inches shorter than the length, the height must be 3 inches B. taller than the width, and the box must have a volume of 56 cubic inches. Let x represent the length in inches. The width is x - 2 and the height is x + 3. ℓwh = V (x )( x - 2) ( x + 3) = 56 x3 + x 2 - 6x = 56 x3 + x 2 - 6x - 56 = 0 Module 7 346 Lesson 1 Finding Rational Solutions of Polynomial Equations 346 ELABORATE C. INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Have students discuss how they could use a graphing utility to help determine which numbers from their list of possible rational zeros are more likely than others to be zeros. Students should recognize that they can use the x-intercepts of the graph to help them focus in on which numbers on their lists are good candidates to test as possible zeros. Use the Rational Root Theorem. Factors of -56: ±1, ±2, ±4, ±7, ±8, ±14, ±28, ±56 Test the possible roots to find one that is actually a root. Use a synthetic division table. m n _ 1 1 -6 -56 1 1 2 -4 -60 2 1 3 0 -56 4 1 5 14 0 Factor the polynomial. using 4 as a root. (x - 4)( x2 + 5x + 14 ) = 0 The quadratic polynomial produces only complex roots. The only possible answer for the D. length is 4 inches. The width is 2 inches and the height is 7 inches. Using a graphing calculator, the graphs intersect at (4, 56), which validates the answer. Elaborate 9. For a polynomial function with integer coefficients, how are the function’s coefficients and rational zeros related? __, The rational zeros of a polynomial function with integer coefficients are in the form m n where m is a factor of the constant term and n is a factor of the leading coefficient. QUESTIONING STRATEGIES If a cubic function has only one rational root, what will be true about the quadratic polynomial quotient that results from synthetic division by the rational root? It will not be factorable over the set of integers. 10. Describe the process for finding the rational zeros of a polynomial function with integer coefficients. Using the Rational Zero Theorem to find all possible rational zeros, test the possible zeros to find one that is actually a zero by using a synthetic division table to organize the work and factor the polynomial. 11. How is the Rational Root Theorem useful when solving a real-world problem about the volume of an object when the volume function is a polynomial and a specific value of the function is given? The theorem is useful in this case because it allows you to find the rational roots of the polynomial equation created when you set the volume function equal to the given value. SUMMARIZE THE LESSON How can you use the Rational Root Theorem to find the rational solutions of a polynomial equation? You can write the equation in the form p( x) = 0, and then use the theorem to identify possible roots of the equation. These roots will be of p the form __ q . You can then test the possible roots using synthetic substitution. If you can reduce the polynomial to a quadratic, you can try factoring the quadratic to find any other rational roots. By rewriting the equation so that one side is 0, you can use the Rational Root Theorem to find the dimension given by the variable and then find the other dimensions. 12. Essential Question Check-In What does the Rational Root Theorem find? The Rational Root Theorem finds the possible rational roots of a polynomial equation. Module 7 347 Lesson 1 LANGUAGE SUPPORT Communicating Math Have students work in pairs. Instruct one student to write a polynomial function in factor form. Have the second student identify the zeros of the function and explain why they are the zeros. The students switch roles and repeat the process. Repeat the example from the lesson to provide a format. 347 Lesson 7.1 EVALUATE Find the rational zeros of each polynomial function. Then write each function in factored form. 1. ƒ( x ) = x 3 − x 2− 10x − 8 2. Factors of −8 : ±1, ±2, ±4, ±8 4 is a zero. 5 is a zero. (x - 4 )(x 2 + 3x + 2) = 0 (x - 4 )(x + 2 ) ( x + 1) = 0 x = 5, x = -3, or x = -4 f( x)= (x - 5 ) (x + 3)( x + 4) f( x) = ( x - 4)( x + 2) (x + 1 ) j( x ) = 2x 3 - x2 - 13x - 6 4. Factors of −15 : ±1, ±3, ±5, ±15 3 is a zero. 1 is a zero. (x - 3 )(2x + 5x + 2) = 0 (x - 1) (x - 8x + 15 ) = 0 (x - 1) (x -5) (x - 3) = 0 2 (x - 3 )(2x + 1 )(x + 2 ) = 0 x = 1, x = 5, or x = 3 g( x) = (x - 1) (x - 5 )(x - 3) 1 x = 3, x = -_ , or x = -2 2 j( x) = ( x - 3) (2x + 1 )(x + 2) h (x ) = x 3 - 5x2 + 2x + 8 6. 2 is a zero. 2 is a zero. 3 (x - 2 )(2x + 1) (3x + 1) = 0 1, or x = - _ 1 x = 2, x = -_ 2 3 h( x) = (x - 2) (2x + 1 )( 3x + 1) s ( x) = x - x − x + 1 8. 3 t (x ) = x + x Factors of 1 : ±1 Factors of −12 : ±1, ±2, ±3, ±4, ±6, ±12 1 is a zero. 3 is a zero. (x - 1 ) (x2 - 1) = 0 (x - 1 ) (x + 1)( x - 1) = 0 (x - 3) (x2 + 4x + 4) = 0 (x - 3) (x + 2) (x + 2 ) = 0 x = 1 or x = -1 x = 3 or x = -2 s( x) = (x - 1 ) ( x + 1)(x - 1 ) t( x) = (x - 3 ) (x + 2)( x + 2) 348 Module 7 Exercise Depth of Knowledge (D.O.K.) Exercise 17 Example 1 Finding Zeros Using the Rational Zero Theorem Exercises 2–12 Example 2 Solving a Real-World Problem Using the Rational Root Theorem Exercises 13–16 INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Students can use patterns in the signs of the (x - 2 )(6x 2 + 5x + 1 ) = 0 2 Practice Explore Relating Zeros and Coefficients of Polynomial Functions 2 Factors of −2 : ±1, ±2 x = 2, x = 4, or x = -1 h (x ) = (x - 2 ) ( x - 4)(x + 1 ) Concepts and Skills h ( x ) = 6x3 - 7x - 9x − 2 Factors of 8 : ±1, ±2, ±4, ±8 (x - 2 )(x 2 - 3x - 4) = 0 (x - 2 )(x - 4 ) ( x + 1) = 0 7. g ( x) = x 3 - 9x2 + 23x − 15 Factors of −6 : ±1, ±2, ±3, ±6 2 5. ASSIGNMENT GUIDE (x - 5) (x2 + 7x + 12 ) = 0 (x - 5) (x + 3) (x + 4 ) = 0 x = 4, x = -2, or x = -1 3. ƒ( x )= x 3 + 2x2- 23x - 60 Factors of −60 : ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60 terms in the polynomial function to help them decide which of the possible rational zeros to test. For example, if the signs of the terms in the polynomial function (or in the quotient after dividing synthetically) are all positive, students need not check any positive numbers on their lists. Lesson 1 COMMON CORE Mathematical Practices 1–12 1 Recall of Information MP.5 Using Tools 13–17 2 Skills/Concepts MP.4 Modeling 18–19 2 Skills/Concepts MP.3 Logic 20 3 Strategic Thinking MP.2 Reasoning 21 3 Strategic Thinking MP.3 Logic Finding Rational Solutions of Polynomial Equations 348 AVOID COMMON ERRORS Students may incorrectly conclude that a polynomial function that has n rational zeros has only n real zeros. Explain that the function may have irrational zeros as well, and irrational zeros are real zeros. 9. k (x ) = x4 + 5x 3 - x 2 − 17x + 12 10. g(x ) = x 4 - 6x 3 + 11x 2 - 6x Factors of 12 : ±1, ±2, ±3, ±4, ±6, ±12 g (x) = x (x3 - 6x 2 + 11x - 6) 1 is a zero. Factors of -6 : ±1, ±2, ±3, ±6 1 is a zero. Factor the polynomial. ( x − 1) (x 3 + 6x2 + 5x − 12 ) (x )( x - 1) (x 2 - 5x + 6) = 0 Factors of -12 : ±1, ±2, ±3, ±4, ±6, ±12 (x )( x - 1) (x - 3 )( x - 2) = 0 x = 1, x = 0, x = 3, or x = 2 1 is a zero. CONNECT VOCABULARY ( x - 1) (x - 1)( x 2 + 7x + 12) = 0 ( x - 1) (x - 1)( x + 3 )( x + 4) = 0 Have students, in their own words, explain how the Rational Zero Theorem and the Rational Root Theorem are related (for example, a solution of a polynomial equation is often called a root). x = 1, x = -3, or x = -4 g (x) = (x ) (x - 1 )( x - 3) (x - 2 ) k(x) = ( x - 1)( x - 1)(x + 3 ) (x + 4) 11. h( x ) = x4 - 2x 3 - 3x 2 + 4x + 4 12. ƒ( x) = x 4 - 5x 2 + 4 Factors of 4 : ±1, ±2, ±4 Factors of 4 : ±1, ±2, ±4 2 is a zero. 1 is a zero. ( x − 2) (x 3 − 3x − 2) f (x ) = (x - 1 )(x 3 + x 2 − 4x − 4) Factors of -2 : ±1, ±2 Factors of −4 : ±1, ±2, ±4 2 is a zero. 2 is a zero. ( x - 2) (x - 2)( x 2 + 2x + 1) = 0 ( x - 2) (x - 2)( x + 1 )( x + 1) = 0 (x - 1 )(x - 2 ) (x 2 + 3x + 2) = 0 x = -1 or x = 2 x = 1, x = 2, x = -2, or x = -1 h(x )= (x - 2) (x - 2 )( x + 1) (x + 1 ) f (x ) = (x - 1 )(x - 2 )(x + 2 )( x + 1 ) (x - 1 )(x - 2 ) (x + 2) (x + 1) = 0 13. Manufacturing A laboratory supply company is designing a new rectangular box in which to ship glass pipes. The company has created a box with a width 2 inches shorter than its length and a height 9 inches taller than twice its length. The volume of each box must be 45 cubic inches. What are the dimensions? Let x represent the length in inches. Then the width is x - 2 and the height is 2x + 9. ℓwh = V (x )( x - 2)( 2x + 9) = 0 2x 3 + 5x 2 - 18x = 45 2x3 + 5x 2 - 18x - 45 = 0 Factors of -45: ±1, ±3, ±5, ±9, ±15, ±45 3 is a root. ( x - 3) ( 2x2 + 11x + 15) = 0, so ( x - 3)( 2x + 5) ( x + 3) = 0 ___, or x = -3 x = 3, x = -5 2 Length cannot be negative. The length must be is 3 inches. The width is 1 inch and the height is 15 inches. Module 7 349 Lesson 7.1 349 Lesson 1 14. Engineering A natural history museum is building a pyramidal glass structure for its tree snake exhibit. Its research team has designed a pyramid with a square base and with a height that is 2 yards more than a side of its base. The volume of the pyramid must be 147 cubic yards. What are the dimensions? CRITICAL THINKING Let x represent the side of the square base in yards. The height is x + 2. _1 ℓwh = V 3 _1( x) (x) (x + 2) = 147 3 1 _(x3 + 2x2) = 147 3 x3 + 2x 2 = 441 x3 + 2x 2 - 441 = 0 7 is a root. Factors of -441: ±1, ±3, ±7, ±9, ±21, ±49, ±63, ±147, ±441 Discuss with students why the Rational Root Theorem works, by applying it to a quadratic equation, such as 2x2 + x - 15 = 0, and showing how the process of solving the equation by factoring focuses on the factors of p and q in a way that is similar to the process of the Rational Root Theorem. Focus students’ attention on how p is the product of the first coefficients of the factors, and q is the product of the constant terms of the factors. ( x - 7) (x 2 + 9x + 63) = 0 The quadratic factor produces only complex roots. So, each side of the base is 7 yards and the height is 9 yards. 15. Engineering A paper company is designing a new, pyramidshaped paperweight. Its development team has decided that to make the length of the paperweight 4 inches less than the height and the width of the paperweight 3inches less than the height. The paperweight must have a volume of 12 cubic inches. What are the dimensions of the paperweight? Let x represent the height in inches. The length is x - 4 and the width is x - 3. _1 ℓwh = V 3 _1( x - 4)(x - 3) ( x) = 12 3 _1(x3 - 7x 2 + 12x ) = 12 3 x 3 - 7x 2 + 12x = 36 x3 - 7x 2 + 12x - 36 = 0 Factors of -36: ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36 6 is a root. (x - 6 ) (x2 - x + 6) = 0 The quadratic factor produces only complex roots. So, the height is 6 inches, the length is 2 inches, and the width is 3 inches. Module 7 350 Lesson 1 Finding Rational Solutions of Polynomial Equations 350 PEER-TO-PEER DISCUSSION Ask students to discuss with a partner how the Rational Root Theorem, in conjunction with the Zero Product Property, enables them to solve real-world problems that can be modeled by polynomial equations. The Rational Root Theorem can be used to identify possible solutions. Identifying one or more of the solutions from the list of possible solutions can help you to write the equation in factored form. You can then use the Zero Product Property to set each factor equal to zero and solve for other possible solutions. 16. Match each set of roots with its polynomial function. B 3 A. x = 2, x = 3, x = 4 ƒ(x) = (x + 2 )(x + 4 ) x – _ 2 C 1 3 5 7 _ _ _ B. x = –2, x = –4, x = x– x +_ ƒ(x) = x – 4 2 2 3 1, x = _ 5 , x = –_7 A C. x = _ ƒ(x) = (x – 2 )( x – 3 ) (x – 4 ) 2 4 3 6 D ƒ(x) = x + _ 4 4 6 ( x–_ D. x = –_ , x = _, x = 4 x – 4) 5 7 5 7 ( ( )( )( ( ) )( ) ) 17. Identify the zeros of ƒ( x) = (x + 3)(x - 4)(x - 3), writethe function in standard form, and state how the zeros are related to thestandard form. The zeros of f (x) are x = -3, x = 4, and x = 3. f (x) =( x + 3)(x - 4) (x - 3) = (x 2 + 3x - 4x - 12 )( x - 3 ) = ( x 2 - x - 12 )( x - 3) = x 3 - 3x 2 - x 2 + 3x - 12x + 36 = x 3 - 4x 2 - 9x + 36 The zeros of f (x) are all factors of the constant term in the polynomial function. H.O.T. Focus on Higher Order Thinking JOURNAL Have students describe how they could use the Rational Zero Theorem to write a polynomial function in intercept form. 18. Critical Thinking Consider the polynomial function g( x) = 2x3 - 6x 2 + πx + 5. Is it possible to use the Rational Zero Theorem and synthetic division to factor this polynomial? Explain. No; it is not possible because the function contains a term, πx, whose coefficient is irrational and, therefore, not an integer. 19. Explain the Error Sabrina was told to find the zeros of the polynomial function h( x ) = x( x - 4) (x + 2 ). She stated that the zeros of this polynomial are x = 0, x = -4, and x = 2. Explain her error. For any factor (ax + b) , a zero occurs at -_ a . Sabrina forgot to include the b negative sign when converting from her factors to the zeros. 20. Justify Reasoning If _c is a rational zero of a polynomial function p( x) , explain why b bx - c must be a factor of the polynomial. c is a factor of p( ) by the Factor Theorem. So, Since p _c = 0, x - _ x b b 1 b p(x ) = x - c_ q (x) and p(x) = _ q(x) =_b (bx - c) q( x) , which x - c_ b b b () ( ) ( ) shows that bx - c is a factor of p( x). 21. Justify Reasoning A polynomial function p(x ) has degree 3, and its zeros are –3, 4, and 6. What do you think is the equation of p( x)? Do you think there could be more than one possibility? Explain. p(x ) = (x + 3)( x - 4 )( x - 6); any constant multiple of p (x ) will also have degree 3 and the same zeros, so the equation can be any function of the form p(x) = a (x + 3 )(x - 4 )( x - 6) where a ≠ 0. Module 7 351 Lesson 7.1 351 Lesson 1 Lesson Performance Task AVOID COMMON ERRORS For the years from 2001–2010, the number of Americans traveling to other countries by plane can be represented by the polynomial function A( t ) = 20t 4 - 428t 3 + 2760t 2 - 4320t + 33,600 , where A is the number of thousands of Americans traveling abroad by airplane and t is the number of years since 2001. In which year were there 40,000,000 Americans traveling abroad? Use the Rational Root Theorem to find your answer. [Hint: consider the function’s domain and range before finding all possible rational roots.] Some students may set A(t) equal to 40,000,000, which is the number given in the problem. Ask students to check the units of A. thousands of Americans. Have students divide 40,000,000 by 1,000 to get the correct value for A, 40,000. More precisely, A is 40,000 thousands of Americans. A(t ) = 20t 4 - 428t 3 + 2760t 2 - 4320t + 33,600 40,000 = 20t 4 - 428t 3 + 2760t 2 - 4320t + 33,600 0 = 20t 4 - 428t 3 + 2760t 2 - 4320t - 6400 QUESTIONING STRATEGIES Factors of -6400 between 0 and 9: 1, 2, 4, 5, 8. Test the possible roots: _ m n 20 -428 2760 -4320 -6400 1 20 -408 2352 -1968 -8368 2 20 -388 1984 -352 -7104 4 20 -348 1368 1152 -1792 5 20 -328 1120 1280 0 8 20 -268 616 608 -1536 Why is it useful to know a function’s domain when solving for the roots? If the domain consists only of rational numbers, then the roots must be rational. For example, if the domain consists of the integers from 0 to 9, then the roots must be rational because integers are rational numbers. 0 = (x - 5) (20x 3 - 328x 2 + 1120x + 1280) Factors of 1280 between 0 and 9: 1, 2, 4, 5, 8. Test the possible roots: _ m n 20 -328 1120 1280 1 20 -308 812 2092 2 20 -288 544 2368 4 20 -248 128 1792 5 20 -228 -20 1180 8 20 -168 -224 -512 Why does the domain consist only of integers? The domain is the number of years since 2001. The function only makes sense for integer values. Because the cubic polynomial factor has no rational roots between 0 and 9, x = 5 years returns the only solution. In other words, there were 40,000,000 Americans traveling overseas by air in 2006. Module 7 352 Lesson 1 EXTENSION ACTIVITY Have students research the factors that affect tourist numbers, such as changes in economic status, or the safety of a destination. Have students discuss who might use a model of tourist numbers like A(t) and how it might be used. Ask students to describe situations in which it would be useful to input a value of t to calculate the number of tourists, and in what situations it would be useful to do the inverse—use a given number of tourists and solve for the roots. 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 Rational Solutions of Polynomial Equations 352