7.1 Finding Rational Solutions of Polynomial Equations
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1 Name Class Date 7.1 Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio? Resource Locker Explore Relatig Zeros ad Coefficiets of Polyomial Fuctios The zeros of a polyomial fuctio ad the coefficiets of the fuctio are related. Cosider the polyomial fuctio ƒ (x) = (x + 2) (x -1) (x + 3). A B Idetify the zeros of the polyomial fuctio. Multiply the factors to write the fuctio i stadard form. C How are the zeros of ƒ (x) related to the stadard form of the fuctio? D E Now cosider the polyomial fuctio g (x) = (2x + 3) (4x - 5) (6x - 1). Idetify the zeros of this fuctio. Multiply the factors to write the fuctio i stadard form. F How are the zeros of g (x) related to the stadard form of the fuctio? Module Lesso 1
2 Reflect 1. I geeral, how are the zeros of a polyomial fuctio related to the fuctio writte i stadard form? 2. Discussio Does the relatioship from the first Reflect questio hold if the zeros are all itegers? Explai. 3. If you use the zeros, you ca write the factored form of g (x) as g (x) = (x )(x )(x ), rather tha as g (x) = (2x + 3) (4x - 5) (6x - 1). What is the relatioship of the factors betwee the two forms? Give this relatioship i a geeral form. Explai 1 Fidig Zeros Usig the Ratioal Zero Theorem If a polyomial fuctio p (x) is equal to ( a 1 x + b 1 ) ( a 2 x + b 2 ) ( a 3 x + b 3 ), where a 1, a 2, a 3, b 1, b 2, ad b 3 are itegers, the leadig coefficiet of p (x) will be the product a 1 a 2 a 3 ad the costat term will be b the product b 1 b 2 b 3. The zeros of p (x) will be the ratioal umbers - 1 a, - b 2 1 a, - b 3 2 a. 3 Comparig the zeros of p (x) to its coefficiet ad costat term shows that the umerators of the polyomial s zeros are factors of the costat term ad the deomiators of the zeros are factors of the leadig coefficiet. This result ca be geeralized as the Ratioal Zero Theorem. Ratioal Zero Theorem If p (x) is a polyomial fuctio with iteger coefficiets, ad if _ m is a zero of p (x) m (p( the m is a factor of the costat term of p (x) ad is a factor of the leadig coefficiet of p(x). ) = 0), Example 1 Fid the ratioal zeros of the polyomial fuctio; the write the fuctio as a product of factors. Make sure to test the possible zeros to fid the actual zeros of the fuctio. A ƒ (x) = x x 2-19x - 20 a. Use the Ratioal Zero Theorem to fid all possible ratioal zeros. Factors of -20: ±1, ±2, ±4, ±5, ±10, ±20 b. Test the possible zeros. Use a sythetic divisio table to orgaize the work. I this table, the first row (shaded) represets the coefficiets of the polyomial, the first colum represets the divisors, ad the last colum represets the remaiders. m_ Module Lesso 1
3 c. Factor the polyomial. The sythetic divisio by 4 results i a remaider of 0, so 4 is a zero ad the polyomial i factored form is give as follows: (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, ad x = -1. B ƒ (x) = x 4-4 x 3-7 x x + 24 a. Use the Ratioal Zero Theorem to fid all possible ratioal zeros. Factors of 24: ±, ±, ±, ±, ±, ±, ±, ± b. Test the possible zeros. Use a sythetic divisio table. m_ c. Factor the polyomial. The sythetic divisio by results i a remaider of 0, so is a zero ad the polyomial i factored form is give as follows: (x - ) ( x 3 - x 2 - x - ) = 0 d. Use the Ratioal Zero Theorem agai to fid all possible ratioal zeros of g (x) = x 3 - x 2 - x -. Factors of -8: ±, ±, ±, ± e. Test the possible zeros. Use a sythetic divisio table. m_ f. Factor the polyomial. The sythetic divisio by results i a remaider of 0, so is a zero ad the polyomial i factored form is: (x - ) (x - ) ( x 2 + x + ) = 0 (x - ) (x - ) (x + ) (x + ) = 0 x =, x =, x =, or x = The zeros are. Module Lesso 1
4 Reflect 4. How is usig sythetic divisio o a 4 th degree polyomial to fid its zeros differet tha usig sythetic divisio o a 3 rd degree polyomial to fid its zeros? 5. Suppose you are tryig to fid the zeros the fuctio ƒ (x) = x Would it be possible to use sythetic divisio o this polyomial? Why or why ot? 6. Usig sythetic divisio, you fid that 1 2 is a zero of ƒ (x) = 2 x 3 + x 2-13x + 6. The quotiet from the sythetic divisio array for ƒ ( 1 2 ) is 2 x 2 + 2x Show how to write the factored form of ƒ (x) = 2 x 3 + x 2-13x + 6 usig iteger coefficiets. Your Tur 7. Fid the zeros of ƒ (x) = x 3-2 x 2-8x. m_ Module Lesso 1
5 Explai 2 Solvig a Real-World Problem Usig the Ratioal Root Theorem Sice a zero of a fuctio ƒ (x) is a value of x for which ƒ (x) = 0, fidig the zeros of a polyomial fuctio p (x) is the same thig as fid the solutios of the polyomial equatio p (x) = 0. Because a solutio of a polyomial equatio is kow as a root, the Ratioal Zero Theorem ca be also expressed as the Ratioal Root Theorem. Ratioal Root Theorem If the polyomial p (x) has iteger coefficiets, the every ratioal root of the polyomial equatio p (x) = 0 ca be writte i the form m, where m is a factor of the costat term of p (x) ad is a factor of the leadig coefficiet of p (x). Example 2 Egieerig A pe compay is desigig a gift cotaier for their ew premium pe. The marketig departmet has desiged a pyramidal box with a rectagular base. The base width is 1 ich shorter tha its base legth ad the height is 3 iches taller tha 3 times the base legth. The volume of the box must be 6 cubic iches. What are the dimesios of the box? Graph the volume fuctio ad the lie y = 6 o a graphig calculator to check your solutio. History i the markig A. Aalyze Iformatio The importat iformatio is that the base width must be ich shorter tha the base legth, the height must be iches taller tha 3 times the base legth, ad the box must have a volume of cubic iches. B. Formulate a Pla Write a equatio to model the volume of the box. Let x represet the base legth i iches. The base width is height is, or. 1_ 3 lw h = V 1_ 3 ( ) (x - ) (3) (x + ) = x 3 - x - = 0 ad the Module Lesso 1
6 C. Solve Use the Ratioal Root Theorem to fid all possible ratioal roots. Factors of -6: ±, ±, ±, ± Test the possible roots. Use a sythetic divisio table. m_ Factor the polyomial. The sythetic divisio by results i a remaider of 0, so is a root ad the polyomial i factored form is as follows: ( x - ) ( x 2 + x + ) = 0 The quadratic polyomial produces oly roots, so the oly possible aswer for the base legth is iches. The base width is ich ad the height is iches. D. Justify ad Evaluate The x-coordiates of the poits where the graphs of two fuctios, f ad g, itersect is the solutio of the equatio f (x) = g (x). Usig a graphig calculator to graph the volume fuctio ad y = 6 results i the graphs itersectig at the poit. Sice the x-coordiate is, the aswer is correct. Your Tur 8. Egieerig A box compay is desigig a ew rectagular gift cotaier. The marketig departmet has desiged a box with a width 2 iches shorter tha its legth ad a height 3 iches taller tha its legth. The volume of the box must be 56 cubic iches. What are the dimesios of the box? Module Lesso 1
7 m_ Elaborate 9. For a polyomial fuctio with iteger coefficiets, how are the fuctio s coefficiets ad ratioal zeros related? 10. Describe the process for fidig the ratioal zeros of a polyomial fuctio with iteger coefficiets. 11. How is the Ratioal Root Theorem useful whe solvig a real-world problem about the volume of a object whe the volume fuctio is a polyomial ad a specific value of the fuctio is give? 12. Essetial Questio Check-I What does the Ratioal Root Theorem fid? Module Lesso 1
8 Evaluate: Homework ad Practice Fid the ratioal zeros of each polyomial fuctio. The write each fuctio i factored form. Olie Homework Hits ad Help Extra Practice 1. ƒ (x) = x 3 x 2 10x 8 2. ƒ (x) = x x 2-23x j (x) = 2 x 3 - x 2-13x g (x) = x 3-9 x x h (x) = x 3-5 x 2 + 2x h (x) = 6 x 3-7 x 2-9x 2 7. s (x) = x 3 - x 2 x t (x) = x 3 + x 2 8x 12 Module Lesso 1
9 9. k (x) = x x 3 - x 2 17x g (x) = x 4-6 x x 2-6x 11. h (x) = x 4-2 x 3-3 x 2 + 4x ƒ (x) = x 4-5 x Maufacturig A laboratory supply compay is desigig a ew rectagular box i which to ship glass pipes. The compay has created a box with a width 2 iches shorter tha its legth ad a height 9 iches taller tha twice its legth. The volume of each box must be 45 cubic iches. What are the dimesios? Module Lesso 1
10 14. Egieerig A atural history museum is buildig a pyramidal glass structure for its tree sake exhibit. Its research team has desiged a pyramid with a square base ad with a height that is 2 yards more tha a side of its base. The volume of the pyramid must be 147 cubic yards. What are the dimesios? 15. Egieerig A paper compay is desigig a ew, pyramidshaped paperweight. Its developmet team has decided that to make the legth of the paperweight 4 iches less tha the height ad the width of the paperweight 3 iches less tha the height. The paperweight must have a volume of 12 cubic iches. What are the dimesios of the paperweight? Image Credits: James Kigma/Shutterstock Module Lesso 1
11 16. Match each set of roots with its polyomial fuctio. A. x = 2, x = 3, x = 4 ƒ (x) = (x + 2) (x + 4) ( x 3_ 2 ) B. x = 2, x = 4, x = 3_ 2 ƒ (x) = ( x 1_ C. x = 1_ 2, x = 5_ 4, x = _ 7 3 D. x = _ 4 5, x = 6_ 7, x = 4 ƒ (x) = ( x + 4_ 2) ( x 5_ 4 ) ( x + ƒ (x) = (x 2) (x 3) (x 4) 5) ( x 6_ 7) 7_ 3 ) (x 4) 17. Idetify the zeros of ƒ (x) = (x + 3) (x - 4) (x - 3), write the fuctio i stadard form, ad state how the zeros are related to the stadard form. H.O.T. Focus o Higher Order Thikig 18. Critical Thikig Cosider the polyomial fuctio g (x) = 2 x 3-6 x 2 + πx + 5. Is it possible to use the Ratioal Zero Theorem ad sythetic divisio to factor this polyomial? Explai. 19. Explai the Error Sabria was told to fid the zeros of the polyomial fuctio h (x) = x (x - 4) (x + 2). She stated that the zeros of this polyomial are x = 0, x = -4, ad x = 2. Explai her error. 20. Justify Reasoig If c_ is a ratioal zero of a polyomial fuctio p (x), explai why b bx - c must be a factor of the polyomial. 21. Justify Reasoig A polyomial fuctio p (x) has degree 3, ad its zeros are 3, 4, ad 6. What do you thik is the equatio of p (x)? Do you thik there could be more tha oe possibility? Explai. Module Lesso 1
12 Lesso Performace Task For the years from , the umber of Americas travelig to other coutries by plae ca be represeted by the polyomial fuctio A (t) = 20 t t t t + 33,600, where A is the umber of thousads of Americas travelig abroad by airplae ad t is the umber of years sice I which year were there 40,000,000 Americas travelig abroad? Use the Ratioal Root Theorem to fid your aswer. [Hit: cosider the fuctio s domai ad rage before fidig all possible ratioal roots.] m_ m_ Image Credits: Paul Seheult/Eye Ubiquitous/Corbis Module Lesso 1
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