7.1 Finding Rational Solutions of Polynomial Equations

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1 4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio? Resource Locker Commo Core Math Stadards The studet is expected to: A-APR. Kow ad apply the Remaider Theorem: For a polyomial p(x) ad a umber a, the remaider o divisio by x a is p(a), so p(a) = 0 if ad oly if (x a) is a factor of p(x). Also A-APR., A-CED. Mathematical Practices MP. Reasoig Laguage Objective Explai to a parter how to idetify the factors of a polyomial fuctio. ENGAGE Essetial Questio: How do you fid the ratioal roots of a polyomial equatio? Use the Ratioal Root Theorem to idetify possible ratioal roots. Check each by usig sythetic substitutio. If a ratioal root is foud, repeat the process o the quotiet obtaied from the bottom row of the sythetic substitutio. Cotiue to fid ratioal roots i this way util the quotiet is quadratic, at which poit you ca try factorig to idetify the last two ratioal roots. 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 + ) (x -) (x + ). A B C D E F Idetify the zeros of the polyomial fuctio. The zeros are x = -, x =, ad x = -. Multiply the factors to write the fuctio i stadard form. f (x) = (x + ) (x - ) (x + ) = ( x + x - x - ) (x + ) = ( x + x - ) (x + ) = x + x + x + x - x - 6 = x + 4 x + x - 6 How are the zeros of ƒ (x) related to the stadard form of the fuctio? Each of the zeros of the polyomial fuctio is a factor of the costat term i the stadard form. Now cosider the polyomial fuctio g (x) = (x + ) (4x - 5) (6x - ). Idetify the zeros of this fuctio. The zeros are x = - _, ad x = _ 4 6., x = 5_ Multiply the factors to write the fuctio i stadard form. g (x) = (x + ) (4x - 5) (6x - ) = (8 x - 0x + x - 5) (6x - ) = (8 x + x - 5) (6x - ) = 48 x - 8 x + x - x - 90x + 5 = 48 x + 4 x - 9x + 5 How are the zeros of g (x) related to the stadard form of the fuctio? Each of the umerators of the zeros is a factor of the costat term, 5, ad each of the deomiators is a factor of the leadig coefficiet, 48. PREVIEW: LESSON PERFORMANCE TASK View the Egage sectio olie. Discuss the photo ad how the umber of tourists i ay give year ca vary depedig o may factors. The preview the Lesso Performace Task. A_MNLESE85894_UM07L.idd 4 Module 7 4 Lesso DO NOT EDIT--Chages must be made through File ifo CorrectioKey=NL-A;CA-A Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio? A-APR. Kow ad apply the Remaider Theorem: For a polyomial p (x) ad a umber a, the remaider o divisio by x - a is p (a), so p (a) = 0 if ad oly if (x - a) is a factor of p (x). Also A-APR., A-CED. 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 + ) (x -)(x + ). Idetify the zeros of the polyomial fuctio. Multiply the factors to write the fuctio i stadard form. f (x) = (x + ) (x - ) (x + ) How are the zeros of ƒ (x) related to the stadard form of the fuctio? = (x + x - x - ) (x + ) = (x + x - ) (x + ) = x + x + x + x - x - 6 Now cosider the polyomial fuctio g (x) = (x + ) (4x - 5) (6x - ). Idetify the zeros of this fuctio. Multiply the factors to write the fuctio i stadard form. = x + 4 x + x - 6 The zeros are x = -, x =, ad x = -. How are the zeros of g (x) related to the stadard form of the fuctio? Resource Each of the zeros of the polyomial fuctio is a factor of the costat term i the stadard form. The zeros are x = - _, x = 5_, ad x = _ g (x) = (x + ) (4x - 5) (6x - ) 6. = (8 x - 0x + x - 5) (6x - ) = (8 x + x - 5) (6x - ) = 48 x - 8 x + x - x - 90x + 5 = 48 x + 4 x - 9x + 5 Each of the umerators of the zeros is a factor of the costat term, 5, ad each of the deomiators is a factor of the leadig coefficiet, 48. Module 7 4 Lesso A_MNLESE85894_UM07L.idd 4 /9/4 :9 PM HARDCOVER PAGES Tur to these pages to fid this lesso i the hardcover studet editio. /9/4 :7 PM 4 Lesso 7.

2 Reflect. I geeral, how are the zeros of a polyomial fuctio related to the fuctio writte i stadard form? Each of the umerators of the zeros is a factor of the costat term. Each of the deomiators of the zeros is a factor of the leadig coefficiet.. Discussio Does the relatioship from the first Reflect questio hold if the zeros are all itegers? Explai. Yes; If the zeros are all itegers, each of them ca be writte with a deomiator of. Each of the umerators is still a factor of the costat term.. If you use the zeros, you ca write the factored form of g (x) as g (x) = (x + )(x )(x - 6 ), rather tha as g (x) = (x + ) (4x - 5) (6x - ). What is the relatioship of the factors betwee the two forms? Give this relatioship i a geeral form. I each factor, the deomiator of the fractio becomes the coefficiet of the variable. I geeral, if the zero is - b_ a, the factor ca be writte as (ax + b). Explai Fidig Zeros Usig the Ratioal Zero Theorem If a polyomial fuctio p (x) is equal to ( a x + b ) ( a x + b ) ( a x + b ), where a, a, a, b, b, ad b are itegers, the leadig coefficiet of p (x) will be the product a a a ad the costat term will be b the product b b b. The zeros of p (x) will be the ratioal umbers - a, - b a, - b a. 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 is a zero of p (x) (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 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. ƒ (x) = x + x - 9x - 0 a. Use the Ratioal Zero Theorem to fid all possible ratioal zeros. Factors of -0: ±, ±, ±4, ±5, ±0, ±0 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 Module 7 4 Lesso PROFESSIONAL DEVELOPMENT EXPLORE Relatig Zeros ad Coefficiets of Polyomial Fuctios INTEGRATE TECHNOLOGY Studets have the optio of completig the Explore activity either i the book or olie. QUESTIONING STRATEGIES What is the relatioship betwee the factors of a polyomial fuctio ad the zeros of the fuctio? The zeros are the values of x foud by settig each factor equal to 0 ad solvig for x. If a zero of a polyomial fuctio is 7, what do you kow about the coefficiets whe the polyomial is writte i stadard form? 7 is a factor of the costat term ad is a factor of the leadig coefficiet. EXPLAIN Fidig Zeros Usig the Ratioal Zero Theorem QUESTIONING STRATEGIES Is every zero of a polyomial fuctio represeted i the set of umbers give by the Ratioal Zero Theorem? No. The Ratioal Zero Theorem gives oly those zeros that are ratioal umbers. A polyomial fuctio ca also have zeros that are irratioal umbers or imagiary umbers. A_MNLESE85894_UM07L 4 Itegrate Mathematical Practices This lesso provides a opportuity to address Mathematical Practice MP., which calls for studets to traslate betwee multiple represetatios ad to reaso abstractly ad quatitatively. Studets explore the relatioship betwee the factors of a polyomial fuctio ad its zeros. They lear how to idetify the zeros give the factors, ad the factors give the zeros. They the explore the relatioships betwee the ratioal zeros of a fuctio ad its leadig coefficiet ad costat term, establishig the Ratioal Zero Theorem. 6/0/4 0:4 AM Fidig Ratioal Solutios of Polyomial Equatios 4

3 AVOID COMMON ERRORS Some studets may forget to iclude ad i their list of possible ratioal zeros. You may wat to suggest that they write these first so that they are ot iadvertetly left off the list. QUESTIONING STRATEGIES If the leadig coefficiet of a polyomial fuctio with iteger coefficiets is, what ca you coclude about the fuctio s ratioal zeros? Explai your reasoig. They must be itegers, because whe you apply the Ratioal Zero Theorem, ca equal oly or i. 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 + 6x + 5) = 0 (x - 4) (x + 5) (x + ) = 0 x = 4, x = -5, or x = - The zeros are x = 4, x = -5, ad x = -. B ƒ (x) = x 4-4 x - 7 x + x + 4 a. Use the Ratioal Zero Theorem to fid all possible ratioal zeros. Factors of 4: ±, ±, ±, ± 4, ± 6, ± 8, ±, ± 4 b. Test the possible zeros. Use a sythetic divisio table 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 - x - 0 x - 8 ) = 0 d. Use the Ratioal Zero Theorem agai to fid all possible ratioal zeros of g (x) = x - x - 0 x - 8. Factors of -8: ±, ±, ± 4, ± e. Test the possible zeros. Use a sythetic divisio table f. 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: (x - )(x - 4 )( x + x + ) = 0 (x - )(x - 4 )(x + )(x + ) = 0 x =, x = 4, x = -, or x = - The zeros are x =, x = 4, x = -, ad x = -. Module 7 4 Lesso A_MNLESE85894_UM07L.idd 4 COLLABORATIVE LEARNING Small Group Activity Have studets work i groups of 4 studets. Istruct each group to create a fifth-degree polyomial fuctio with ratioal zeros, ot all of which are itegers. Ask them to write their fuctios i stadard from. Have groups exchage fuctios, ad have each group create a poster showig how to apply the Ratioal Zero Theorem to fid the zeros of the fuctio. Studets posters should also show verificatio that each umber is ideed a zero of the fuctio. 7/7/4 0:4 AM 4 Lesso 7.

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 rd degree polyomial to fid its zeros? To fid the zeros of a 4 th degree polyomial usig sythetic divisio, you eed to use sythetic divisio to reduce that polyomial to a rd degree polyomial ad the use sythetic divisio agai to reduce that polyomial to a quadratic polyomial that ca be factored, if possible. 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? It would ot be possible to fid the zeros of this polyomial usig sythetic substitutio because the fuctio has o ratioal roots, oly complex roots. 6. Usig sythetic divisio, you fid that is a zero of ƒ (x) = x + x - x + 6. The quotiet from the sythetic divisio array for ƒ ( ) is x + x -. Show how to write the factored form of ƒ (x) = x + x - x + 6 usig iteger coefficiets. Usig as a zero ad the quotiet x + x - you ca write f (x) = x + x - x + 6 as f (x) = ( x - _ ) ( x + x - ). f (x) = ( x - _ ) ( x + x - ) = ( x - _ ) () ( x + x - 6) = (x - ) ( x + x - 6) = (x - ) (x + ) (x - ) INTEGRATE MATHEMATICAL PRACTICES Focus o Math Coectios MP. Remid studets that a zero of a fuctio is a umber from the domai that the fuctio pairs with 0. Discuss that, for this reaso, a graph of the fuctio will have a x-itercept at each zero. Studets ca the make a cocrete coectio betwee the ratioal zeros they idetify for a fuctio, ad the role the zeros play i the graph of the fuctio. Your Tur 7. Fid the zeros of ƒ (x) = x + x - x- 5. a. Use the Ratioal Zero Theorem. Factors of -5: ±, ±, ±5, ±5 b. Test the possible zeros to fid oe that is actually a zero c. Factor the polyomial usig as a zero. (x - ) ( x + 6x + 5) = 0 (x - ) (x + ) (x + 5) = 0 x =, x = -, or x = -5 The zeros are x =, x = -, ad x = -5. Module 7 44 Lesso DIFFERENTIATE INSTRUCTION Visual Cues A_MNLESE85894_UM07L.idd 44 Ecourage studets to circle the leadig coefficiet i the fuctio ad to write is a factor of above it, ad to circle the costat term i the fuctio ad to write m is a factor of above it. This will be helpful whe applyig the Ratioal Zero Theorem, ad will keep studets from erroeously writig the reciprocals of the possible ratioal zeros, especially sice the usages of m ad appear i reverse alphabetical order with respect to the fuctio. 0/6/4 : PM Fidig Ratioal Solutios of Polyomial Equatios 44

5 EXPLAIN Solvig a Real-World Problem Usig the Ratioal Root Theorem CONNECT VOCABULARY Explai how the words zeros ad roots (or solutios) have similar meaigs but are used i differet cotexts. The zeros of a fuctio are the roots (or solutios) of the related equatio. QUESTIONING STRATEGIES Why is it ecessary to rewrite the equatio so that it is equal to 0? I order to fid the roots of a equatio usig the Ratioal Root Theorem, the equatio must be i the form p (x) = 0. What iformatio is obtaied by applyig the Ratioal Zero Theorem to a polyomial fuctio? A list of all possible ratioal zeros of the fuctio Explai 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 _, where m is a factor of the costat term of p (x) ad is a factor of the leadig coefficiet of p (x). 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 ich shorter tha its base legth ad the height is iches taller tha 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. 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 times the base legth, ad the box must have a volume of 6 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 x +, or (x + ). lw h = V ( x )(x - )()(x + ) = 6 x - x - 6 = 0 x - ad the History i the markig Module 7 45 Lesso A_MNLESE85894_UM07L 45 6/8/4 : PM 45 Lesso 7.

6 C. Solve Use the Ratioal Root Theorem to fid all possible ratioal roots. Factors of -6: ±, ±, ±, ± 6 Test the possible roots. Use a sythetic divisio table 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 + x + ) = 0 The quadratic polyomial produces oly complex roots, so the oly possible aswer for the base legth is iches. The base width is ich ad the height is 9 iches. INTEGRATE MATHEMATICAL PRACTICES Focus o Critical Thikig MP. Prompt studets to recogize that ay ratioal roots foud by factorig the resultig quadratic polyomial must be umbers that were idetified as possible ratioal roots iitially. This may help them to catch errors i factorig, or i performig the sythetic divisio. 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 (, 6). 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 iches shorter tha its legth ad a height iches taller tha its legth. The volume of the box must be 56 cubic iches. What are the dimesios of the box? A. The box width must be iches shorter tha the legth, the height must be iches taller tha the width, ad the box must have a volume of 56 cubic iches. B. Let x represet the legth i iches. The width is x - ad the height is x +. lwh = V (x) (x - ) (x + ) = 56 x + x - 6x = 56 x + x - 6x - 56 = 0 Module 7 46 Lesso A_MNLESE85894_UM07L 46 6/7/4 0: PM Fidig Ratioal Solutios of Polyomial Equatios 46

7 ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus o Techology MP.5 Have studets discuss how they could use a graphig utility to help determie which umbers from their list of possible ratioal zeros are more likely tha others to be zeros. Studets should recogize that they ca use the x-itercepts of the graph to help them focus i o which umbers o their lists are good cadidates to test as possible zeros. QUESTIONING STRATEGIES If a cubic fuctio has oly oe ratioal root, what will be true about the quadratic polyomial quotiet that results from sythetic divisio by the ratioal root? It will ot be factorable over the set of itegers. SUMMARIZE THE LESSON How ca you use the Ratioal Root Theorem to fid the ratioal solutios of a polyomial equatio? You ca write the equatio i the form p (x) = 0, ad the use the theorem to idetify possible roots of the equatio. These roots will be of the form p q. You ca the test the possible roots usig sythetic substitutio. If you ca reduce the polyomial to a quadratic, you ca try factorig the quadratic to fid ay other ratioal roots. C. Use the Ratioal Root Theorem. Factors of -56: ±, ±, ±4, ±7, ±8, ±4, ±8, ±56 Test the possible roots to fid oe that is actually a root. Use a sythetic divisio table. Elaborate p_ q Factor the polyomial. usig 4 as a root. (x - 4) ( x + 5x + 4) = 0 The quadratic polyomial produces oly complex roots. The oly possible aswer for the legth is 4 iches. The width is iches ad the height is 7 iches. D. Usig a graphig calculator, the graphs itersect at (4, 56), which validates the aswer. 9. For a polyomial fuctio with iteger coefficiets, how are the fuctio s coefficiets ad ratioal zeros related? The ratioal zeros of a polyomial fuctio with iteger coefficiets are i the form _, where m is a factor of the costat term ad is a factor of the leadig coefficiet. 0. Describe the process for fidig the ratioal zeros of a polyomial fuctio with iteger coefficiets. Usig the Ratioal Zero Theorem to fid all possible ratioal zeros, test the possible zeros to fid oe that is actually a zero by usig a sythetic divisio table to orgaize the work ad factor the polyomial.. 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? The theorem is useful i this case because it allows you to fid the ratioal roots of the polyomial equatio created whe you set the volume fuctio equal to the give value. By rewritig the equatio so that oe side is 0, you ca use the Ratioal Root Theorem to fid the dimesio give by the variable ad the fid the other dimesios.. Essetial Questio Check-I What does the Ratioal Root Theorem fid? The Ratioal Root Theorem fids the possible ratioal roots of a polyomial equatio. Module 7 47 Lesso A_MNLESE85894_UM07L 47 LANGUAGE SUPPORT Commuicatig Math Have studets work i pairs. Istruct oe studet to write a polyomial fuctio i factor form. Have the secod studet idetify the zeros of the fuctio ad explai why they are the zeros. The studets switch roles ad repeat the process. Repeat the example from the lesso to provide a format. 6/7/4 0: PM 47 Lesso 7.

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 EVALUATE. ƒ (x) = x x 0x 8. ƒ (x) = x + x - x - 60 Factors of 8 : ±, ±, ±4, ±8 Factors of 60 : ±, ±, ±, ±4, ±5, ±6, 4 is a zero. ±0, ±, ±5, ±0, ±0, ±60 5 is a zero. (x - 4) ( x + x + ) = 0 (x - 4) (x + ) (x + ) = 0 x = 4, x = -, or x = - f (x) = (x - 4) (x + ) (x + ) (x - 5) ( x + 7x + ) = 0 (x - 5) (x + ) (x + 4) = 0 x = 5, x = -, or x = -4 f (x) = (x - 5) (x + ) (x + 4). j (x) = x - x - x g (x) = x - 9 x + x 5 Factors of 6 : ±, ±, ±, ±6 is a zero. is a zero. (x - ) ( x + 5x + ) = 0 (x - ) (x + ) (x + ) = 0 x =, x = - _, or x = - j (x) = (x - ) (x + ) (x + ) Factors of 5 : ±, ±, ±5, ±5 (x - ) ( x - 8x + 5) = 0 (x - ) (x -5) (x - ) = 0 x =, x = 5, or x = g (x) = (x - ) (x - 5) (x - ) ASSIGNMENT GUIDE Cocepts ad Skills Explore Relatig Zeros ad Coefficiets of Polyomial Fuctios Example Fidig Zeros Usig the Ratioal Zero Theorem Example Solvig a Real-World Problem Usig the Ratioal Root Theorem Practice Exercise 7 Exercises Exercises 6 5. h (x) = x - 5 x + x h (x) = 6 x - 7 x - 9x Factors of 8 : ±, ±, ±4, ±8 is a zero. is a zero. (x - ) ( x - x - 4) = 0 (x - ) (x - 4) (x + ) = 0 x =, x = 4, or x = - h (x) = (x - ) (x - 4) (x + ) Factors of : ±, ± (x - ) ( 6x + 5x + ) = 0 (x - ) (x + ) (x + ) = 0 x =, x = - _, or x = - _ h (x) = (x - ) (x + ) (x + ) 7. s (x) = x - x x + 8. t (x) = x + x 8x Factors of : ± Factors of : ±, ±, ±, ±4, ±6, ± is a zero. is a zero. (x - ) ( x - ) = 0 (x - ) (x + ) (x - ) = 0 x = or x = - s (x) = (x - ) (x + ) (x - ) (x - ) ( x + 4x + 4) = 0 (x - ) (x + ) (x + ) = 0 x = or x = - t (x) = (x - ) (x + ) (x + ) INTEGRATE MATHEMATICAL PRACTICES Focus o Patters MP.8 Studets ca use patters i the sigs of the terms i the polyomial fuctio to help them decide which of the possible ratioal zeros to test. For example, if the sigs of the terms i the polyomial fuctio (or i the quotiet after dividig sythetically) are all positive, studets eed ot check ay positive umbers o their lists. Module 7 48 Lesso A_MNLESE85894_UM07L 48 Exercise Depth of Kowledge (D.O.K.) Mathematical Practices 6/0/4 0:7 AM Recall of Iformatio MP.5 Usig Tools 7 Skills/Cocepts MP.4 Modelig 8 9 Skills/Cocepts MP. Logic 0 Strategic Thikig MP. Reasoig Strategic Thikig MP. Logic Fidig Ratioal Solutios of Polyomial Equatios 48

9 AVOID COMMON ERRORS Studets may icorrectly coclude that a polyomial fuctio that has ratioal zeros has oly real zeros. Explai that the fuctio may have irratioal zeros as well, ad irratioal zeros are real zeros. CONNECT VOCABULARY Have studets, i their ow words, explai how the Ratioal Zero Theorem ad the Ratioal Root Theorem are related (for example, a solutio of a polyomial equatio is ofte called a root). 9. k (x) = x x - x 7x + 0. g (x) = x 4-6 x + x - 6x Factors of : ±, ±, ±, ±4, ±6, ± is a zero. Factor the polyomial. (x ) ( x + 6 x + 5x ) Factors of : ±, ±, ±, ±4, ±6, ± is a zero. (x - ) (x - ) ( x + 7x + ) = 0 (x - ) (x - ) (x + ) (x + 4) = 0 x =, x = -, or x = -4 k (x) = (x - ) (x - ) (x + ) (x + 4) g (x) = x ( x - 6 x + x - 6) Factors of -6 : ±, ±, ±, ±6 is a zero. (x) (x - ) ( x - 5x + 6) = 0 (x) (x - ) (x - ) (x - ) = 0 x =, x = 0, x =, or x = g (x) = (x) (x - ) (x - ) (x - ). h (x) = x 4 - x - x + 4x + 4. ƒ (x) = x 4-5 x + 4 Factors of 4 : ±, ±, ±4 is a zero. is a zero. (x ) ( x x ) Factors of - : ±, ± is a zero. is a zero. Factors of 4 : ±, ±, ±4 f (x) = (x - ) ( x + x 4x 4) Factors of 4 : ±, ±, ±4 (x - ) (x - ) ( x + x + ) = 0 (x - ) (x - ) (x + ) (x + ) = 0 x = - or x = h (x) = (x - ) (x - ) (x + ) (x + ) (x - ) (x - ) ( x + x + ) = 0 (x - ) (x - ) (x + ) (x + ) = 0 x =, x =, x = -, or x = - f (x) = (x - ) (x - ) (x + ) (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 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? Let x represet the legth i iches. The the width is x - ad the height is x + 9. lwh = V (x) (x - ) (x + 9) = 0 x + 5 x - 8x = 45 x + 5 x - 8x - 45 = 0 Factors of -45: ±, ±, ±5, ±9, ±5, ±45 is a root. (x - ) ( x + x + 5) = 0, so (x - ) (x + 5) (x + ) = 0 x =, x = -5, or x = - Legth caot be egative. The legth must be is iches. The width is ich ad the height is 5 iches. Module 7 49 Lesso A_MNLESE85894_UM07L 49 6/0/4 0:46 AM 49 Lesso 7.

10 4. 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 yards more tha a side of its base. The volume of the pyramid must be 47 cubic yards. What are the dimesios? Let x represet the side of the square base i yards. The height is x +. _ lwh = V _ (x) (x)(x + ) = 47 _ ( x + x ) = 47 x + x = 44 x + x - 44 = 0 Factors of -44: ±, ±, ±7, ±9, ±, ±49, ±6, ±47, ±44 7 is a root. (x - 7) ( x + 9x + 6) = 0 The quadratic factor produces oly complex roots. So, each side of the base is 7 yards ad the height is 9 yards. CRITICAL THINKING Discuss with studets why the Ratioal Root Theorem works, by applyig it to a quadratic equatio, such as x + x - 5 = 0, ad showig how the process of solvig the equatio by factorig focuses o the factors of p ad q i a way that is similar to the process of the Ratioal Root Theorem. Focus studets attetio o how p is the product of the first coefficiets of the factors, ad q is the product of the costat terms of the factors. 5. 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 iches less tha the height. The paperweight must have a volume of cubic iches. What are the dimesios of the paperweight? Let x represet the height i iches. The legth is x - 4 ad the width is x -. _ lwh = V _ (x - 4) (x - ) (x) = _ ( x - 7x + x) = x - 7 x + x = 6 x - 7 x + x - 6 = 0 Factors of -6: ±, ±, ±, ±4, ±6, ±9, ±, ±8, ±6 6 is a root. (x - 6) ( x - x + 6) = 0 The quadratic factor produces oly complex roots. So, the height is 6 iches, the legth is iches, ad the width is iches. Image Credits: James Kigma/Shutterstock Module 7 50 Lesso A_MNLESE85894_UM07L.idd 50 /9/4 :7 PM Fidig Ratioal Solutios of Polyomial Equatios 50

11 PEER-TO-PEER DISCUSSION Ask studets to discuss with a parter how the Ratioal Root Theorem, i cojuctio with the Zero Product Property, eables them to solve real-world problems that ca be modeled by polyomial equatios. The Ratioal Root Theorem ca be used to idetify possible solutios. Idetifyig oe or more of the solutios from the list of possible solutios ca help you to write the equatio i factored form. You ca the use the Zero Product Property to set each factor equal to zero ad solve for other possible solutios. 6. Match each set of roots with its polyomial fuctio. B A. x =, x =, x = 4 ƒ (x) = (x + ) (x + 4) ( x _ B. x =, x = 4, x = _ C. x = _, x = _ 5 4, x = _ 7 D. x = 4_ 5, x = _ 6 C A ƒ (x) = ( x ) _ ( x _ 5 ) 4) ( x + _ 7 ) ƒ (x) = (x ) (x ) (x 4) 7, x = 4 D ƒ (x) = ( x + 4_ 5) ( x _ 7) 6 (x 4) 7. Idetify the zeros of ƒ (x) = (x + ) (x - 4) (x - ), write the fuctio i stadard form, ad state how the zeros are related to the stadard form. The zeros of f (x) are x = -, x = 4, ad x =. f (x) = (x + ) (x - 4) (x - ) = ( x + x - 4x - ) (x - ) = ( x - x - ) (x - ) = x - x - x + x - x + 6 = x - 4 x - 9x + 6 The zeros of f (x) are all factors of the costat term i the polyomial fuctio. JOURNAL Have studets describe how they could use the Ratioal Zero Theorem to write a polyomial fuctio i itercept form. H.O.T. Focus o Higher Order Thikig 8. Critical Thikig Cosider the polyomial fuctio g (x) = x - 6 x + πx + 5. Is it possible to use the Ratioal Zero Theorem ad sythetic divisio to factor this polyomial? Explai. No; it is ot possible because the fuctio cotais a term, πx, whose coefficiet is irratioal ad, therefore, ot a iteger. 9. Explai the Error Sabria was told to fid the zeros of the polyomial fuctio h (x) = x (x - 4) (x + ). She stated that the zeros of this polyomial are x = 0, x = -4, ad x =. Explai her error. For ay factor (ax + b), a zero occurs at - b_ a. Sabria forgot to iclude the egative sig whe covertig from her factors to the zeros. 0. 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. Sice p ( c_ b) = 0, x - c_ is a factor of p (x) by the Factor Theorem. So, b p (x) = ( x - b) c_ q (x) ad p (x) = b_ b( x - b) c_ q (x) = _ (bx - c) q (x), which b shows that bx - c is a factor of p (x).. Justify Reasoig A polyomial fuctio p (x) has degree, ad its zeros are, 4, ad 6. What do you thik is the equatio of p (x)? Do you thik there could be more tha oe possibility? Explai. p (x) = (x + ) (x - 4) (x - 6) ; ay costat multiple of p (x) will also have degree ad the same zeros, so the equatio ca be ay fuctio of the form p (x) = a (x + ) (x - 4) (x - 6) where a 0. Module 7 5 Lesso A_MNLESE85894_UM07L 5 6/0/4 0:5 AM 5 Lesso 7.

12 Lesso Performace Task For the years from 00 00, the umber of Americas travelig to other coutries by plae ca be represeted by the polyomial fuctio A (t) = 0 t 4-48 t t - 40t +,600, where A is the umber of thousads of Americas travelig abroad by airplae ad t is the umber of years sice 00. 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.] A (t) = 0 t 4-48 t t - 40t +,600 40,000 = 0 t 4-48 t t - 40t +,600 0 = 0 t 4-48 t t - 40t Factors of betwee 0 ad 9:,, 4, 5, 8. Test the possible roots: (x - 5) (0 x - 8 x + 0x + 80) Factors of 80 betwee 0 ad 9:,, 4, 5, 8. Test the possible roots: Because the cubic polyomial factor has o ratioal roots betwee 0 ad 9, x = 5 years returs the oly solutio. I other words, there were 40,000,000 Americas travelig overseas by air i 006. Image Credits: Paul Seheult/Eye Ubiquitous/Corbis AVOID COMMON ERRORS Some studets may set A (t) equal to 40,000,000, which is the umber give i the problem. Ask studets to check the uits of A. thousads of Americas. Have studets divide 40,000,000 by,000 to get the correct value for A, 40,000. More precisely, A is 40,000 thousads of Americas. QUESTIONING STRATEGIES Why is it useful to kow a fuctio s domai whe solvig for the roots? If the domai cosists oly of ratioal umbers, the the roots must be ratioal. For example, if the domai cosists of the itegers from 0 to 9, the the roots must be ratioal because itegers are ratioal umbers. Why does the domai cosist oly of itegers? The domai is the umber of years sice 00. The fuctio oly makes sese for iteger values. Module 7 5 Lesso EXTENSION ACTIVITY A_MNLESE85894_UM07L 5 Have studets research the factors that affect tourist umbers, such as chages i ecoomic status, or the safety of a destiatio. Have studets discuss who might use a model of tourist umbers like A (t) ad how it might be used. Ask studets to describe situatios i which it would be useful to iput a value of t to calculate the umber of tourists, ad i what situatios it would be useful to do the iverse use a give umber of tourists ad solve for the roots. 6/9/5 :5 AM Scorig Rubric poits: Studet correctly solves the problem ad explais his/her reasoig. poit: Studet shows good uderstadig of the problem but does ot fully solve or explai his/her reasoig. 0 poits: Studet does ot demostrate uderstadig of the problem. Fidig Ratioal Solutios of Polyomial Equatios 5

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