3.4 Iania Beemon

%/éé ? Math 110 Guided Lecture Sheet Sect 3.4 A - i flicients Rational Roots Theorem: If the polynomial P(z) = anz™ + ap_12" 1+ o+ @a1T + o has integer coc ( an # 0 and aq # 0), then every rational zero of P is of the form where 170 rcte e 176) oF atl POk p +- P @ Zes oOF Yhe e £ /0/(; where p and q are integers and ' Z) UseS Sy 4 HiC S 1o cvedluily g P 15 a factor of the constant coefficient ag /Oo/j(]m'(/// o caety el gr Ll COrSBAl q is a factor of the leading cocflicient ay,. S . N /0 Fech Koy Ma QN Zeos oy 1 Lo, ohere, h M = S e ; (han ) e Feprean Wrr /S 200 A« q F T e T o st % a= ( p 3)/ ) sty //C//(/ Z *r 77 /2/0/5 wH f ./ / s 400 //"* l'/ sy , 3 / /' Lol ra /¢ T SRl P isti tential 1. Find the real zeros of the following polynomials and write the polynomial if factor_et.i ?OM- Start by listing the pote: rational zeros. Find at least one zero, and use this to find the other factors by division. P(z) =23 +222 - 132+ 10 - edl ruclers oF 107 1 i | pewtss 10 Lt zJi%f/Maj D A 151 e e ) possels Zeas oF #1¢ fcmf/d«,' Aw//fl loe o] M + I > -lo o ~ (OCIJ:O - ~7,%S, /o Véfl o065 or LX) ¢ /00()' ()2;/) /—5 1= Fa{,/r_/"z afi/oa) =7 0 -y (xFl)= P XY = x-1) cx S+ Sx-foy O =x¢t/ _ PE) =438 A K4S ) 2 gag =g (P6F) T CX1) Cars, Cx-2) L/ / 3 0 - p(l):o—> x=1 18 ave g Fhc 12—47'&& B I B ) i ) . ) / & ;a4 Ofpéxjcf(xlaab\a sckse plx) | (x). = (F1Y(x2 b 9 x+4) = | I kg O lgga)—,/x—ljéxm/w—z) ' { -7 = - i , 055 16(¢ ratdmd z.oe =FhR=V) 2g Hrr A (0 KFZ 29 xo mL c:1 4=t "F2 29 4o 4 (e =% pilifa,ty Pl s 2 W ) T, Y -2t 2,4

(. _ _fg_)7(z+x7fiv)£('d A xc-oig?-—x '7_""54/' "ThTe 7079 3 =X SHi- Z:x%fa' 4 X _'_L, _ /=7 /=9 P/:,, ;= Y ¢- ©OF )= A SrFF T (x7d ~0 Hpomy (x20 w Sorr oy €0z =X 44X 7 31 CT+ X7 Yo o |- 11 X Fo X ) (L4 R 7 (7707717 @z s v T 1- —- /I\ b (/%= € x)d S 2% & 0:(2\707 ~<=7 71 71- N a e z- (€ / @) ()()fi/ L ) B d Y S f-T X e- (== (1~ - | ¢ ! | (wod # € - - C. 1 | (¢ Ayrrr Sy (1-x) m (X107 _p Doz g5 ) =X C—O'—(/?d (x 77/ MA ST Jru /ot 0q1% sd/ st | 202 g:(,pwa/ G+ TE— oI — 7T + T = (2)d (?—)«7((%7«75/—7%7(4 v <><7fl/ (o-x)(r-, %4108 4X TAt X kakald cux)/~Istx) 0 €~ /- Tioh w—xi/ L-¢- 3¢ 1, A s A (€-x-, %2/ 1 1k F31H bo tc- o h ¢ z f o n lz (] ¢ b A — b 5 e -l a ) e T e o T br $4 T+ L+ 1T 5,637 > J 6+ ;T8 — 2y = (2)g DL VI NS F / 722,

= 4 = il = M i ke e Wl = 2. (Review 3. 2) Find all the real zeros and sketch a graph of the polynomial. P (. t P(I) T'3I"—41'+12 /\ .; ay) JT] I -{ ' 10 b N\ */_V k 2\ .) /\ 7/ ') \J,L\ Vs ( J 8 N\ L f_ 7 b ;' v 47§ L +/ 4 \ /, L+ /9\) //1/ ") G 2 - U' DBy S 0\ t \ IS p 3 ,G/ 1 0 1 ;;/ 5'7'/ + LO /b \\(' \ - / | 4 I) } Oy # \/V 3. (Review from 3.2) Use the graph of the sixth degree polynomial p(x) below to answer the following. a. List cach zero of f in point form, and state its likely multiplicity (kecp in mind this is a 6th degree polynomial). b. State the y-intercept in point form. c. Write a possible formula for p(z). You can leave this in factored form. Remember to use your y-intercept to find a, the leading cocfficient. /O/K)'a(K'LQ)Z[)(—/) 2(x-3) me{ofi) (0-13% /0~ >) ° 8 4 8| 5| 4 3| o (49 Z _Ire P PO S S ()(J—Q]Z (r-1)( #- %) W":,; (ra |5 M- Frx=3)