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lesson_06 - Pa 9 35 LESSon 6 3 Read Sections 3.2 3.34...

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Unformatted text preview: Pa 9 35 LESSon 6 3 Read Sections 3.2, 3.34 ESFQCWLj , Exames 6,7 3’“? oj 89:11:?“ 5.2 and 5H Hue exam‘ahs in Section 3.3. §e€tion 31,- 1I 7, 9, ll}, 15, 15) 17, 1?,21, 33 Section 3.3, 1. 3; 5! 7, $617, :3 11,25 que 36 3-2 Potflnmial Fandrons A P0113 numcal 0—} degree TL , where n [‘5 o. nonejutl‘v’f’ {htxajrerI has the form (Ruhr amw m+a.-’\c+ao where om, 0i,~‘-,0n are Constanrsi and (“+0. The numbers no, 05%“ are called bhe coefiici‘em hi the Pal/lynching an F3 fieleaclrnj OOijLI‘CiPn‘fI and an {3 He ennstqm term . Some QXMplns 03L [Delanommb are Shawn in the tab{&. misxfiws 25 O ‘XW—P .XIO Graphs 0-}- gcme basic Pohjnomfd funcflons 1:1 1’} 5 y ) 357$ 3 ygrx ) X 7? For a {Janna-WM PCXYDT dejfeg n, PM) "Qn'xn+Qn—r7‘n-L~'~fQ-X+Q =Q 78(14 91’ —L a“ J— 1‘ . .__. +~--+ I 0 1'1 an {X + Cu: 0(2- Chaim 4‘ Clo 1 ) H [M b ‘ ‘ an ’X" A3 ‘9‘“) 95 0T X QCcmea’ (‘2er m mqfihlh‘d9) €H'h421 Positive"? (wrffren Gs 'x—aDo) a nejah‘vetj [wrffl'en G5 'X—yhbo)) {419 any”; El , 716—; 323:. and—L. \ J IX” an approach Zero. 30 as (X hemmes Large m Ma3n6+ude fie b-E’hGWOV 03L Pm) (S Mocleleo! by the behavfov of ““9- ‘EEWM 3:6?an Page 37 The behavicv of Ehe \j—UqLuPS o-j He Paws @1th graph of a {JuLjfleh’h'al PC“) :thn+ “n4 ”(Hav- +a. 'X + Clo 'W Is Called H19 end behaviour 9f the (ijh Cm‘C‘O Pam—9-00 as 0:400 Pm~+m as 'x—aJ‘“ FREQ-“9 ~50 as 'x._; 00 PCXJ—aoo as W—a-Do Pom—we 0.3 weave 1 PC0044» as 'x-ewbo zeros of Functions A Number C Is Called (A zero 0'1 Hue function-f [31 3((C}=O. Multiplffifj of a zero If a ImL‘jncmid Jim), when fade/ed completely; MS 6 POW 0f Ht? for”? ('X-Cfr Jim 3 [aoBHNe integer k, He" VIC {3 a "zero of f of MHH‘FMIQ k A zero of muH'iPhCit‘j i (3 Called a Simlahz zero E1. Let fm: yflaxflzfi Fomw comrfiatétjf 13m: minim-+2) = ‘xztwfl‘ml zeros 1 0 "I *2 ‘mu l tilali6:"f‘j 30:0 ['3 a zero of f of MuHilufacrfj 2' 31mins; zeros [Lm'bh mgiflfotfl i) a H- 919 multiPHcf’r K of zero (‘3 even, the” the graph flattens Md 1431‘ touches the yam m eczc ‘ . Ii Hue multiplicity kfiof H9 2% {3 odd, the graph flafiens ahd then Cvosm‘g 'X=——[ and 0(2—2 are the (Rt-Hrs cf X: C _ Page 33 ‘The ngh 0f 0 Pct‘jnmm Simply Crosses Hue ‘x—am‘s ovt simpu zero 1E3: Far Hue Polfinmrql five) :: X4+3ka+1%1:%1(x+1)('x+2)v when X {3 (M99 in magniwde ! its end beham‘w ', laws 0 ._f _ 2 L5 Modded bj 812%? so 'mUHiP’fcfij 2 i 1 hasafi?orh\ ”/ W hat hofsfmns in HI‘Q“h}(clol£:Qh [s cfetevmrned by Fae Zeros and 5 {flew multilififcitt‘es ._ if EX . Grabh the lyoijomfd amzumgfmflf ed\ “1:45 Cambininj theinanmc-YM 9< '_ 50365185 we ubfm‘n I“ HE (31mph Shaun Oh the W?“ \NhE’n ‘X [3 (“’39 (n mafim'fucie! the Qfic-l behavlm of (3m) {3 mode( by 3:4)(5‘ The items of (30;) are 2 (whirlfcl‘fij 5) and ). Thus the cjmiph of a flattens and {2th Chasm; _.] (muHi‘nlfCH‘j 2 Md Hve 31'th flafien‘: and Just teaches the Wom- HHE “Pf-axis on? 2’ 0t _1 The graph (‘5 Shem cm the YE?“- (Here we dmw the ‘X and 11-3393 {7‘ diff-eye?“- SCOLIE’S ) 25.3 _F[nc[;nj Factfifj and ~ZQYDS OT POLleomiaLS The Division M :31;th Suwofl DCX) and FOX) are {afi‘yncmfnps with DCW)#O_ T119“ H19"? 8W5?" “MEMO PULjnmllflflS QCX) th ROE), NEG??? RC‘X‘) {3 Edi?! O 01" has (:19ij (933 than HM? dgargg 0.}. D“), wch Hurt PM): Q[X)-D'(x)+}2f'>(); 5v quivalf’hflj; PM") 2—.- (32m) + BOX) DOV) Dfx). QM) E5 Called fihe Bach?“ when P“) [‘5 Ciiu’fié’d by DR), RM} {5 Called .—-—-—-—'—'_"'_"" the Temaandev . Page 3? E_ Fa'nd Heguofs‘en't @J‘X) and reminder Rm when PCX3239C4+29C3~X+2 "3 divided by 13(59): (Kim-I. SAM“: To keep the diUISIOH erjanized . {hm-t the mgssinj SCI—term a..N P___,__. aft—4% +1! ‘xlww 13x4+2rx3+owtx+z 366—? (x): 398 «a 3%4+ éxiaxl BXW‘xflzx—l) =3'x"+5x3,3x2 ./“___w______,_ Wm “Maw—w (“WP 93: —4'x \‘ —) —4x3—-3x1+4w . "WPXLflX-U: 45(3— SWIM? H‘X“‘—5'><+2 m??— %*:H 2 .9 11 Tammi} H [TAH'XHU =Il")( +221 wit '7 _______’__,_,.’—’—- ‘27‘X+23 éecjwee 01L ~17'X-f-13 :3 1 J which is (9.35 mm H19 dejfez of xl+zx~| [i Thus! ‘che qu’cien‘f &[x):3’x2—4%+~H. 2 and the vemmder Rhoz—fl‘xmsv 711g mmPMWQIM be exlfleshzd as W?" affixi‘xu = [Ma—4w“) (116+sz ~27-7r+23> 6N Q "( _ Bum 9111(in 5X4+2X3JK+2 r: 3%1—4’X+H + £33;— ‘Xfi-flH ‘Xizrx—l = 3*xi4fx+n _. 21W” ’X—i—z‘X—I W.- If the PDLjHOMiM PM) is divided by the Linear fad-cf 3(4) Hlen the reminder ['3 Pan The {mew (31W X—C is a jam: of the Polynemt‘qr Pm) i1“ and my if Pcmzo Paje 40 Ex. 91:“th [DC‘chzxawxiz oomptetfly . 500mm .. We by to guess some zeros by {Vi-“(*W-QWWI usinj er-fcxcfo“! theorem. . Note Pu)=1?’+1‘—2 :0. The fat-W Haeorem {MFI'LQS 0‘“ {‘3 0‘ jaw” of PM)- To find fhe reminihj jfiaCWS, Carrj out H19 Lonj 'Xl4-2rx +2 division on then‘yht . We 0191“" og—n Ixfi’kfio'x ~9— Ptx): '>c3+Xl—2 : WANT—WW) 763—08 APPLU("3 the Euqdmhc Tommie we gun mm 3% Q Uqfion ocl+zx+zzof We jznd 2W2 3 2w __ —2 ila’UH‘z H —2:J.—4 . O (X a w ‘“ "‘3‘ m =5? L :34 2“? where i: f: 22' F‘ ii \ The ZQFOS 0-f— mam-+2319 [maggnarfl number-g} 30 If hag, no Veal jacf‘GYG 133th {MW theorem, Helm: H1“? facwizamn P('><):’X5+x‘12 :(‘PDUX‘iz'X—a) {‘3 eomffefe The Touwmj Yesult tells us whaf rational Number: mid be zeros oT‘gome [3013”th and (s VeYfl {12.2qu fm’ us to 314955 SOME zeros of a WWW-a9 The Raribnat Zero Test Smpom 7% :‘S a rafional ZQYO OT P'C'X) -= OnXh+QhJXhibu ferry-+00 Luheve Cm, “nan are (ntejers and 0n=P=O_ ThEn P dt'mdPS ()0 and g de93 an The Rafiond 28% Test chfides a list 01 ‘JOSSELUP VOW Ex .Detefmm? 0-H Hue laossibitm‘m jaw rational zeros 3 3_ - H l 3% 3568+7x~4~o SoLuTt'w a ('(ecxr the jmcficm‘s; we have BXaFS'Xl-H‘FX-“SZO, This eguwalenf’ QEuatim has 0” Wefiicr‘em’s (ntejefi. 33 mmtiomfl zero but} if —% {3, ava'ttamo 26m: then FclwiclPS (—8“) Edm'cfes 3 Possibdr‘es 1h“ p; i\,i2,14—. :8 , Possibilirs'os Tw @_ it, 13, So a“ bk? Possibimféd in mh‘o‘nap zeroj-E- .aFE . ii, i2, tea—3:3} 33—1. (1%; 1%; :1: 2 3 "3* Page 4: E7: Fac-I‘or H18 PanHDWaI P(%)=3X3~37€)-5%+é OomFlE’tztj 30mm“ We first make a guess of Some ratiend zeros _ By the rational mere {251; if —E is a Emmi) 29”), P E3 Q CEEWSONJL 6» %"3 & dc‘uisov 01L 3_ Thus, \josxlbiiiti‘es 3%: F: ii, in, i3; i6 lmsflhiisfies Tm %; ilf t3_ J PossLbill‘t-‘ses jm J1. :x, $1,151,? :6! 1—1-1: 3-3. W H—fi—E Nou ii 2 it it.% 2 i2 went? 03¢qu {h the Lisf 30 We don’t Yewm‘é’ 9‘9"!- 3 I r‘ To TM a rational zero we '6)” Some : 3x14“ Jr 6 ’ I 0w %m3~%x1~5(x-+6 Sin—whim was just. hots: P(u):3—3_5 338+“: +6 2: “(i-#0, 1 (S 310130 2.9“}. ‘ "HXl-S'Y Pc—1)=~3(~I)3— M-lf—BHHS ', ~—H 95—ch ________,,__ :_5Hg+5+6:0,30 4m awe? n+6 Blj Hue TMTW theorem five—l): 9H1 ' LL . f I 8 dime-hm bn t‘nevijht. \Jf Thus . we obtain. 3X3—8X2—59H6 : (9w!) (axfiuwré). NOBL‘ we can TMW 3Xl—Il‘X-Fé =(3Y~l)(‘X—3> The Tinal answer is 3x3_gx’_.5rx+g '7: (mafiaxuzx'xa) {s q acf'w WECaWj wf {the dfw‘sim Renwlc _ From the Jtclcf'mizflt'm; we see the“ twfiem are -1, 35» 3, In His EWS‘YW‘ . Han are a“ Tatiana? numbers Tke [t‘m‘ We obfained above [ab-fled with (a?) 1‘3 He MST of ai[ )aSSEMQ candidafkfi 1-9 a mfimafi zero: PM there we 0T CfiWSQ MW‘WHG‘M where a ncljnmmfi QE‘uaTiw RM horatimofl zero, aflkmfih that? {5 am? Oj‘fifandiclqtgs‘“ ...
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