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# Note for MATH 1310 at UH

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This 13 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15

Secl39nn u FindingAIIZemsufPulynnm39nls Page I at IS Finding All Zeros of Polynomials In ndlng all zeros ofapolynomlal P faetorthe polynomlal eompletely Ifa faetor res appears ktlmes m the eomplete faetorlzatloh of P0 then we say that is a zero of multlplleltyk Apolynomlal of degree n 2 1 has exaetly h zeroes eouhtmg multrphertres For apolynomlal wrth real eoemerehts complex zeros oeeur m complex eomugate parrs that rs lfthe complex number ahx ls a zero ofP theh rts complex eomugate aehx ls also a zero ofP If ls a zem ofthe polynomlal P theh ls a rant ofthe equatroh Px 0 Example 1 Faetor the polynomlal Px lor eompletely and nd all rts zeros State the multrpherty of eaeh zero Snlutinn tn Example 1 The degree of the polynomlal ls 3 Couhtmg multrphertres the polynomlal has exaetly 3 zeros To nd all zeros ofthe polyhomral factor eompletely wore Tu fasturx2 16 PO xr x ltmlghtbehelpful tu and 2 xxz16 the zemsufx 15 rst xx43741 Thls altematemethuu appearslater The zeros are 0431 auol 4 Eaeh zero ls of multrphertyl NoTETo faetorr2 16 rtmrght be helpful to fmolthe zeros of x2 16 rst The zeros are the roots of the equauoh ns 0 Secl39nn on Finding All Zems anuIynnm39nls P2g22uf13 gt3160 x2167160716 716 x 716 x4 Smce 741 s azero ofx216x741x41 rs afactorofx216 Smce 41x5 azero owns x741 rs afaetor owns So16x4xxi4x Example 2 Let P0 96x22x12 Fmd all roots ofthe equatroh Px 0 Snllltinn tn Example 2 The roots othe equauon Px 0 are the zeros of the polynomial P0 x36x22x12 The degree ofthe polyhormal rs three Couhtmg multrphertres the polyhormal has exaetly 3 zeros To nd all zeros ofthe polyhormal factor eorhpletely s 2 P0 risk rw Why 3 9 were Tu faeterr t z x x62x6 rtrhrghthehelpmlte nd 7 2 7x6x 2 Lhez usufxz2 rst z6zJExer x Thxs alternate methud appearslata39 Thezeros are forfo and x The roots ofthe equatroh x36x22x12 0 are x x 21 6xr xand Secl39nn u FindingAIlZemsufPulynnm39nls rzgezor IS NOTE Tofaetor x2 2 rt rmght be helpful to nd the zeros ofr2 2 rst The zeros are the roots of the equatror x2 2 0 220 44721172 272 XZi XZi IEI Smee 7J5 ls azero ofx22x77 xxEx ls afactorofx22 Smee JE ls azero of x22 X JEI ls afactorofx22 So2 x xx7 x Example 3 Fmd apolynomlal of degree 5 wrth mteger eoemerents that has zeros 51 and x Solution to Example 3 Smee 51 ls azero so ls rts complex conjugate is Also srrlee J3 ls azero so ls rts complex conjugate 745 Thepolynomlalls oftheform Pxax7x751x51x7 1x 3r me re jlresrlmsrmre r r re yruzsuruz arer 28r275 ax5 xo28x3714x275xigj Secl39nn u FindingAIlZemsufPulynnm39nls Page 1 Sam 2 to make all eoemerems rmegers P0 297 2894 75 2x57x 56x3728x2150x775 Example 4 Factor me polynormal Px a ezrz x completely and nd an as zeros State the muluplxmty of each zero Snlutinn m Example 4 The degree of me polynoman rs 3 Counting mumplrerues me polynomra1 has 3 zeros Factor me polynomal completely P0 972 xx2 7 2x 1 xx712 x rs afactor aner appears once m are complete factorization 0 rs a zero of mumplreny 1 x71 rs afactor andlt appears tvnce m are complete factonzauon 1 rs a zero ofmuluphmty 2 Secl39nn u FindingAIIZemsufPulynnm39nls ragesunz Example 5 Faetor the polyhorma1 Px a 4 9x36 eorhp1ete1y and nd a11rts zeros State the rhu1trpherty of eaeh zero Snllltinn tn Example 5 The degree of the po1yhorhra1 15 3 Couhtmg rhu1trphertres the po1yhorhra1 has 3 zeros Faetor the polynomal eorhp1ete1y omre aeterr2 9 rt Pa Msam rmerhymm mIghtbehelpful Lu and the zems efr 9 rst Thxs alternate methud appears1ater x20 4 9x4 M4001 9 r4r3rre 3x x415 afaetor andrt appears phee m the eorhp1ete factonzanon 74 15 a zero ofmuhrpherty 1 x31 15 afaetor andrt appears phee m the eorhp1ete factonzauon 7 15 a zero ofmuhrpherty 1 x731 15 afaetor andrt appears phee m the eorhp1ete factonzauon 31 15 a zero ofmuhrpherty 1 Secl39nn u FindingAIlZemsufPulynnm39nls Pzge uf NOTETo faetor x2 9 rt mrght be helpful to nd the zeros of x2 9 rst The zeros are the roots of the equatroh x2 9 0 290 x2979079 strtee e3 15 azero ofx29x7731 X31 15 afaetor ofx29 strtee 3t 15 azero ofx29x73x 15 afaetor ofx29 5o 8 9 x31x7 3x and Px x4x2 9 r4r3rre3r Example For PO x2 eAr 5 nd all zeros grve the multrpherty of each zero and ther wnte the polynomial m factored form Snllltinn tn Example 5 The degree of the polynomial rs 2 Couhtmg mulhpherhes the polynomial has 2 zeros The zeros ofthe polynomial are the roots of the quadrahe equauon x2 e 4x5 0 Solve the equauon by the quadrahe formula Subshtute a 1 b 74 and c 5 rhto the formula Secl39nn u Finding All Zems ufPuIynnm39nls 75 NE 7 4a 2a 74 rHY r 06 21 4l1se 20 The zeros are 2 and 27 Each zero rs ofmuluplxmty 1 Smee 2 1 rs a zero re 2 1 rs a factor othe polynomial Smee 2 er rs a zero re 2 er rs a factor ofthe polynomial P0 x274x5 re2rre 271 rezerxrez Example 7 Fmd apolynomxal of degree three Wth mteger eoemerems thathas zeros 3and 81 3 Snllltinn m Example 7 Fig 7 quS Secl39nn u FindingAIlZemsufPulynnm39nls Pige ufll Smee 81 15 azero so 15 its complex conjugate 781 2 2 pg 15 afactor ofthe polynomial smee e 15 azero kg 15 afactor ofthe polynomial smee 81 15 azero x81 15 afactor ofthe polynomial smee 781 15 azero Thepolynomxal xs oftheformPxax7 x78xx81 We ean set a 3 to make all the coef cientsmtegers 7 3x372x2192x7128 Px3x37 x264x7128 Example 8 Fmd a polynomial of degree four thh integer coef cients that has Zaros land 431 wk 1 azero ofmulnphmty 2 Snlutinn m Example 8 Secl39nn u FindingAIIZemsufPulynnm39nls P2ge uf3 slncc 4 31 a zero so ls its compch conjugate 4 731 x71 appears twlce as afactor omc polynomlal smcc lls azero ofmultlpllclty 2 x7 4 31 ls a factor ofthe polynomlal smcc 4 31 a zero x4441 ls afactor ofthe polynomlal smcc 473x ls azero Thepolynomlalls oftheform P0 xilfx7431x74731 POax712x7431x74731 l x7431 a We can set a 1 to make all the coef clents lntegers P0 ll42x2 758x 25 Examplc 9 Factor the polynomlal P0 x4 7 256 lnto llnear and lneduclble quadratlc factors wlth real coef clents and mcn factorthe polynomlal completelylnto llnear factors wlth compch coef clents Snlutinn m Examplc 9 Secl39nn u Finding AIIZuus ufPulynnm39nls A 2 2 2 x 256x 16 Factor me dAerrence oftwo squares P0 456 216x116 x2 16 42 Factor me dAerrence oftwo squares P0 x4 256 x2 16x1 16 x4x4x2 16 WWW 1mm 1mm rrreemm quadratic Factor X 16 P0 456 x216x116 xe4x4x116 r4r4r41r41 6 hnurfanmsthhcamplzx cad cxems NOTETo factor x2 161tmaybe helpful 16 nd the zeros rst that 15 nd the roots ofthe equauor x2 16 0 Fig In anS 5min u Finding AIIZuus ufPuIynnm39nls Fig 11 quS xZHSO Ens 045 946 Now gm m facxoxs of 9 16 max correspond to m zeros 11 azcrom 41 Y4HS afacwv 113251 0 stafaclax Exercise Set 43 Roots of Polynomials For each of the polynomials below a Find the zeros of the polynomial and state the multiplicity of each zero b Factor the polynomial completely Note In some problems it may be easier to nd the zeros rst and in 0th ers it may be easier to factor rst 10 11 12 13 14 15 16 17 18 19 20 21 22 Pxx2714x 49 Pxx2 78x16 Px x2 79 Poo 7x2 7 25 Pm 7x2 16 Poo 7x2 100 Poo 4x2 7 25 Pm 9x2 74 Poo 36x2 7 49 Px8lx2 16 Px x6 5x4 Px x5 77x3 Pxx2 72x6 Pcc2 2x5 Pxx2 8x25 Pcc2 74x7 Poo 7x4 6x2 7 27 Pm x4 3x2 728 Pm x4 716 Pm 7x4 781 Px x3 76x2 16x796 Pxx35x2x5 Find a polynomial with integer coefficients that satisfies the given conditions 23 Degree ofpolynomial 2 Zeros Si 75139 24 Degree ofpolynomial 2 Zeros 3i 73139 25 Degree ofpolynomial 2 Zero 7i 26 Degree ofpolynomial 2 Zero 79139 27 Degree ofpolynomial 3 Zeros 0 72139 28 Degree ofpolynomial 3 Zeros 0 6i 29 Degree ofpolynomial 3 Zeros 4i 30 Degree ofpolynomial 3 Zeros 72 3i 31 Degree ofpolynomial 3 Zeros 0 43i 32 Degree ofpolynomial 3 Zeros 0 377i 33 Degree ofpolynomial 3 Zeros 2173139 34 Degree ofpolynomial 3 Zeros 71 4i 35 Degree ofpolynomial 4 Zeros 3i 75139 4 36 Degree of polynomial Zeros i 78139 37 Degree ofpolynomial 4 Zeros 3 multiplicity 2 27139 38 Degree ofpolynomial 4 Zeros 4 multiplicity 2 3i 39 Degree ofpolynomial 3 Zeros 1 74139 Constant Coefficient 64 40 Degree ofpolynomial 3 Zeros 75 i Constant Coefficient 25 Exercise Set 43 Roots of Polynomials For each of the following polynomials a Factor the polynomial into linear and irreducible quadratic factors with real coefficients and then b Factor the polynomial completely into linear factors with complex coef cients 41 Px4x312x29x27 42 Px9x3745x216x780 43 Pxx4725 44 Pxx47100 45 Pxx417x216 46 Pg x4 l3x236

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