The Fundamental Theorem of Algebra. Objective To use the Fundamental Theorem of Algebra to solve polynomial equations with complex solutions

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1 5-6 The Fundamental Theorem of Algebra Content Standards N.CN.7 Solve quadratc equatons wth real coeffcents that have comple solutons. N.CN.8 Etend polnomal denttes to the comple numbers. Also N.CN.9, A.APR.3 Objectve To use the Fundamental Theorem of Algebra to solve polnomal equatons wth comple solutons The frst graph shows the three comple-number solutons of The second graph shows the s solutons of How man comple number solutons does 5 0 have? What are the? How can the patterns ou see help ou fnd the soluton? MATHEMATICAL PRACTICES Lesson Vocabular Fundamental Theorem of Algebra You can factor an polnomal of degree n nto n lnear factors, but sometmes the factors wll nvolve magnar numbers. Essental Understandng The degree of a polnomal equaton tells ou how man roots the equaton has. It s eas to see graphcall that ever polnomal functon of degree has a sngle zero, the -ntercept. However, there appear to be three possbltes for polnomals of degree 2. The correspond to these three graphs: 2 4 O O O Two real zeros One real zero No real zeros Lesson 5-6 The Fundamental Theorem of Algebra 39 ma2se_cc_0506.ndd 39 3/29/ :

2 However, b factorng, ou can see that each related equaton has two roots. 4 5 ( )( 2) 5 0 two real roots, 2 and ( 2 )( 2 ) 5 0 a root of multplct two at ( 2 (2 ))( 2 ( )) 5 0 two comple roots, 2 and Ever quadratc polnomal equaton has two roots, ever cubc polnomal equaton has three roots, and so on. Ths result s related to the Fundamental Theorem of Algebra. The German mathematcan Carl Fredrch Gauss ( ) s credted wth provng ths theorem. Theorem The Fundamental Theorem of Algebra If P () s a polnomal of degree n $, then P () 5 0 has eactl n roots, ncludng multple and comple roots. Problem Usng the Fundamental Theorem of Algebra What are all the roots of ? The polnomal equaton has degree 5. There are 5 roots. The zeros of the functon Use the Ratonal Root and Factor Theorems, snthetc dvson, and factorng. Step The polnomal s n standard form. The possble ratonal roots are 4, 42, 44. Step 2 Evaluate the related polnomal functon for 5. Snce P() 5 0, s a root and 2 s a factor. Use snthetc dvson to factor out 2 : How man lnear factors wll there be? If there are fve roots, there must be fve lnear factors. Step Contnue factorng untl ou have fve lnear factors ( 2 )A B Step 4 The roots are, 2, 22,, and 2. 5 ( 2 )A 4BA 2 B 5 ( 2 )( )( 2)( 2 )( ) B the Fundamental Theorem of Algebra, these are the onl roots. Got It?. What are all the roots of the equaton ? 320 Chapter 5 Polnomals and Polnomal Functons ma2se_cc_0506.ndd 320 3/29/ :

3 Problem 2 Fndng All the Zeros of a Polnomal Functon What are the zeros of f () ? Step Use a graphng calculator to fnd an real roots. The graph of shows real zeros at 523 and 5 3. Does the graph show all of the real roots? Yes; the graphs of quartc functons have one or three turnng ponts. Snce the graph shows three turnng ponts, t wll not turn agan to cross the -as a thrd tme. Step 2 Zero X= 3 Y=0 Zero X=3 Y=0 Factor out the lnear factors 3 and 2 3. Use snthetc dvson twce ( 3)A 3 2 6B 5 ( 3)( 2 3)A 2 2B Step 3 Use the Quadratc Formula. Fnd the comple roots of a 5, b 5, c 5 2 Identf the values of a, b, and c. 2 4 " 4()(2) 2() Substtute. 24!27 2 Smplf. The comple roots are 2!7!7 2 and 2. 2!7!7 Step 4 The four zeros of the functon are 23, 3, 2, and 2. B the Fundamental Theorem of Algebra, there can be no other zeros. Got It? 2. a. What are all the zeros of the functon g() ? b. Reasonng The graph of f () s shown at the rght. 8. Use the turnng ponts to eplan wh the graph does NOT show all of the real zeros of the functon.. The graph of g() 5 f () 4 s a translaton of the graph of f up 4 unts. How man real zeros of g wll the graph of g show? Eplan. 2 O Lesson 5-6 The Fundamental Theorem of Algebra 32 ma2se_cc_0506.ndd 32 3/29/ :

4 Concept Summar The Fundamental Theorem of Algebra Here are equvalent was to state the Fundamental Theorem of Algebra. You can use an one of these statements to prove the others. Ever polnomal equaton of degree n $ has eactl n roots, ncludng multple and comple roots. Ever polnomal of degree n $ has n lnear factors. Ever polnomal functon of degree n $ has at least one comple zero. Lesson Check Do ou know HOW? Fnd the number of roots for each equaton Fnd all the zeros for each functon Do ou UNDERSTAND? MATHEMATICAL PRACTICES 5. Vocabular Gven a polnomal equaton of degree n, eplan how ou determne the number of roots of the equaton. 6. Open-Ended Wrte a polnomal functon of degree 4 wth ratonal coeffcents and two comple zeros of multplct Wrtng Descrbe when to use snthetc dvson and when to use the Quadratc Formula to determne the lnear factors of a polnomal. Practce and Problem-Solvng Eercses MATHEMATICAL PRACTICES A Practce Wthout usng a calculator, fnd all the roots of each equaton. See Problem Fnd all the zeros of each functon. See Problem f () g() f () Chapter 5 Polnomals and Polnomal Functons ma2se_cc_0506.ndd 322 3/29/ :

5 B Appl For each equaton, state the number of comple roots, the possble number of real roots, and the possble ratonal roots Fnd all the zeros of each functon f () g() Thnk About a Plan A polnomal functon, f () , s used to model a new roller coaster secton. The loadng zone wll be placed at one of the zeros. The functon has a zero at 5. What are the possble locatons for the loadng zone? Can ou determne how man zeros ou need to fnd? How can ou use polnomal dvson? What other methods can be helpful? STEM 39. Brdges A twst n a rver can be modeled b the functon f () , 23 # # 2. A ct wants to buld a road that goes drectl along the -as. How man brdges would t have to buld? 40. Error Analss Maurce sas: Ever lnear functon has eactl one zero. It follows from the Fundamental Theorem of Algebra. Cherl dsagrees. What about the lnear functon 5 2? she asks. Its graph s a lne, but t has no -ntercept. Whose reasonng s ncorrect? Where s the flaw? Determne whether each of the followng statement s alwas, sometmes, or never true. 4. A polnomal functon wth real coeffcent has real zeros. 42. Polnomal functons wth comple coeffcents have one comple zero. 43. A polnomal functon that does not ntercept the -as has comple roots onl. 44. Reasonng A 4th degree polnomal functon has zeros at 3 and 5 2. Can 4 also be a zero of the functon? Eplan our reasonng. 45. Open-Ended Wrte a polnomal functon that has four possble ratonal zeros but no actual ratonal zeros. 46. Reasonng Show that the Fundamental Theorem of Algebra must be true for all quadratc polnomal functons. Lesson 5-6 The Fundamental Theorem of Algebra 323 ma2se_cc_0506.ndd 323 3/29/ :

6 C Challenge 47. Use the Fundamental Theorem of Algebra and the Conjugate Root Theorem to show that an odd degree polnomal equaton wth real coeffcents has at least one real root. 48. Reasonng What s the mamum number of ponts of ntersecton between the graphs of a quartc and a quntc polnomal functon? 49. Reasonng What s the least possble degree of a polnomal wth ratonal coeffcents, leadng coeffcent, constant term 5, and zeros at!2 and!3? Show that such a polnomal has a ratonal zero and ndcate ths zero. Standardzed Test Prep SAT/ACT 50. How man roots does f () have? Whch translaton takes 5 u 2 u 2 to 5 u u 2? 2 unts rght, 3 unts down 2 unts left, 3 unts up 2 unts rght, 3 unts up 2 unts left, 3 unts down 52. What s the factored form of the epresson ? 2 ( 2 )( 2) 2 ( )( ) 2 ( )( 2) 2 ( 2 )( ) Short Response 53. How would ou test whether (2, 22) s a soluton of the sstem? e,22 3 $ 2 4 Med Revew 54. Fnd a fourth-degree polnomal equaton wth real coeffcents that has 2 and 23 as roots. See Lesson 5-5. Solve each equaton usng the Quadratc Formula A 2 2B 5 3 See Lesson 4-7. Determne whether a quadratc model ests for each set of values. If so, wrte the model. See Lesson f (2) 5 0, f (2) 5 3, f () f (24) 5, f (25) 5 5, f (26) 5 3 Get Read! To prepare for Lesson 5-7, do Eercses Wrte each polnomal n standard form. 60. ( ) 3 6. ( 2 3) ( ) ( 2 ) ( 5) (4 2 ) 3 See Lesson Chapter 5 Polnomals and Polnomal Functons ma2se_cc_0506.ndd 324 3/29/ :