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# Intermediate Algebra MA 113

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This 0 page Class Notes was uploaded by Fiona Sipes on Sunday November 1, 2015. The Class Notes belongs to MA 113 at Indiana University Purdue University - Fort Wayne taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/233538/ma-113-indiana-university-purdue-university-fort-wayne in Mathematics (M) at Indiana University Purdue University - Fort Wayne.

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Date Created: 11/01/15

Linear Factors and Zeros of Polynomials If fx x2 7 8x 15 x73x 75then x73 andxis are factors of x If We set x equal to zero then We have 1 fx0 9 x278x150 3 lot Plotz Plat x73x750 l x7300rx750 4 x3orx5 3 l 2 1 We say thatx 3 andx 5 are zeros of x X 1 1 2 A E 7 3 9 1 IfWe graph x it has xintercepts at 3 0 and 5 0 71 V 2 The following statements are equivalent 1 xaisazeroofthe 1nction 2 x a is a solution root to the polynomial equation x 0 3 xiais afactorof x 4 a 0 is an xintercept of the mph of x Example 1 Given fx 3xx 7x78 The factors of x are 3x x 7 x 78 The zeros of x are x 0 x 77 x 8 The gmph of fx 3xx 7x78 crosses the xaxis at 00 770 and 80 P1ot1 Hotz r1ot3 H 1ESXC Example 2 Given fx2x773x 1 The factors of xare 2x77and3x1 Thezerosof x are 2x770 gtx 3x10 x The graph of x crosses the xraxis at 35 0 and 703310 Hal Mom Phu VIE2X3 1 X V V2 Vz VH V5 V5 2533333 Examples EU Findthe zeros of fxx34x2712x 3 2 l l Mom Phu Solution x 4x 712x0 iaXA34xz12x xx24x 7120 X W 397 3953 xx6x720 Q g g5 x0 or x60 or x720 Q 53 E x0 or x76 or x2 a 3 e 397 Thezerosof xare x0 76and2 a g X2 Example4 Find apolynornial with zeros x 4 and a 1 42xr439 afactor x 71gtx1is afactor fx x 74x 1 x2 73 r4 Nute The answer is not unique Any nonzero multiple of fx also has these zeros such as gx 5x2715x7 20 ExampleS Fmd apolynomxal Wth zerosx 0 3 and 3 Soluuor x 0 2 x7 0 rs a factor or more srmply x rs a factor kgjzzrr rs afactor 2 x W 29r2 rs afactor 9 x x4x r39r 2 8 719x76 3 9x276 n ths answer rs not umque W xxr x also has these zeros but does not have mce rmeger roemrrems Example 5 Fmd 21th3 xrmtercepts of fx 4x3 400x 4xx2 r25 4xx 5rr 75 and 5 and 5 0 r 0 5 0 mu Plotz not Example 7 0 and 710 Solutlons The zeros af x are x The facto Let x 4andx71 rs af x are x74 and xl X4X1 gt373 74 Cheek The graph af x 73x 74 crosses the xraxls amm and 710 m lou Plotz not I E 39r E x m A a1 9 2 a 2 0 m 78 E A 2 m s s Natax 8 73x 74 also has xrmtercepts at 4 0 and 71 0 7x 8x 1610 fmal the zeros ofthe fuhehah Example 8 Use the graph af x Saluu ah 2n mu Plotz not VI EXAS7X 2 8X1 Nr39 Nr39 Nr39 Nr39 Nr39 75 0 and 71 0 and 71 Nate Thls fuhenah 15 alsa another saluuah to Example 7 Exercises 1 Find all the real zeros of the function and then list all the x intercepts of the function a f x 7x2 x 4x 9 b gx3x2x 8x 5 c hx x2 4 d fx x2 4 e gx 9x2 64 f hxx2 8x 9 g fxx24x 21 h fxx3 3x2 10x i gx3x2 2x 8 j gx 6x2 5xl k hx 4x3 2x2 6x 1 hx 4x3 36x 2 Find a polynomial with integer coef cents with the given zeros and then use a graphing calculator to check your answer a x 0 x 3 b x l x 4 c x0x 6 x6 d x 2 x 4 3 e x 2 x Z 4 5 Note The Department of Mathematical Sciences Web Site httpwwwipfwedumath maintains an Assistance with Graphing Calculators Web Page at httpwwwipfwedumathgraphcalchtml to help you obtain and use a graphing calculator In particular there is a link to a page of Graphing Calculator Instructions providing stepby step help for common commands window set up nding x intercepts using a table etc for many models of graphing calculators

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