1 MrsA. lCgaetbarlda oI Chapter 10 Notes S 1 2 3 4 h.... oW DMMritrvuu iQabcilltd.tt..eu iiMe S ppSid zq qllu4o yyuulwx ta (ai²(3rpnrx i‐xil ²n nty9 h+g‐ig n eb 3 a5ag y) t)ns ²h a(2u 8ra xsbedx ute+ dr‐b 1ape3tc)dra. tat ebtceditnre bnodism nb foiioanmrlo i( maal i +(aa lb ‐b)by ) an added binomial (a‐b)(a+b) 2 10.1 Prime and Composite Numbers What is a factor of a number?_______________________________________
W W A1 Ap A r napchhoynoraad dminttum uaaII_cpss_rrmte _o ee1‐o bn3 s_0ttfei _uhha it_prneeme _ prt c_ ffneir_baaam_giunccem_eett m.rb oore osierrb s nrss s fe auamcoornmocff ati m12isoblnl 0?rpeaet e?ern_ro d_? g_ st_i _ehi_ni_t_n_r_ate___ etn_l?__oa_g _ _i re__ot_g__sr_n__e e_l__lr_aly__f_ r__t ._ p__hg_ _re_a_i_rn_m _ t 1 he atnhnua 1mt htbhaeasrt sn c. o a T noh tbihsee i rws fcra_aict lttleoe ndrs ap bsre itsmhiede e s f Ln8tf Tt ofe aaaorxte oheccccghptteattet’ee ooos pon.ter nrr hrl r a bosif ieie z sam e ozntr a ic o aek t etftw ssgati 2 o a o i stec erw o ton t wts 1 no sh1 h t2o 6o ea r e.6he f8 f e ey Ir .r i 2t1d 1 s.4e tt T i6 h6o iI isp hstn.a 2 gt oueittnh es so n 1opa ts ci 2t6toi qto bo t asu xnchl i reSea ttve4ehi.o ne fv peno a 1ne2b rnorp6t eif xitdem rao = wpi a2cn4m ectr2 ra xo hicifeml traa 2 .titsfc nneeo a xto n tc obc wef2t rae aog1 rin saezrfi6 a trsar2b eec .tao e it xrftof oa efh 2naorc e1e .tuxn od ,pnI 2u2rf rdi, ms 4inwx m bi,t b8ne2oyee, 1 li 2ormofn6a r oicxu.c s rk t2 l2Set otao.ioa,rpt .smpsm lT2i yunoe haixtgf nkat o 1i8gvetfn 6 , ait aagsw.lh n u ftteAeayhehs lc eseoelut es tfospae ehitrrr nhe tei hgme aet 3 Do a factor tree for 924 as shown another way in the book. There are some tricks to figuring out if a number is a factor of a larger one. 1. Every multiple of 2 ends in ___. Example ____ 2. Every multiple of 3 has digits that add up to a multiple of 3. Ex. _______ 3. Every multiple of 5 ends in a ____ or ____. Ex. ________ 4. Every multiple of 10 ends in a ________. Ex _________ The greatest common factor of a set of numbers is the largest integer that is a _________ of all the numbers. If they only have the factor 1 in common they are said to be _____________________. Find the greatest common factor (GCF) of 28 and 63. Make a tree and factor each into primes. 28 63 4 7 9 7 2 2 3 3 2² x 7 3² x 7 The GCF is 7 Look at example 2, p. 447. In that case more than one prime factor is shared by both numbers. The GCF is the product of all the factors in common. Which for 30 and 75 is 5 and 3 so the GCF is 5 x 3 = 15. Example: p. 448, set 2, #8f, h 4 10.2 Monomials and their Factors If we want to factor a monomial we factor the coefficient v ariab elegs. a Tsh fea cptroirms.e factors of 12x² are 2 • 2 • 3 • x • x and list all the W Wf achet aoctar ianenrg fge.i n ttFhhdiene tm dhp e rta higmnerd eeG a tfChtaeFecs ntoto fmcr 1osu 2molxtfmi² 1p ao5lynnxid ynf ³ag1?c 5 at _xol_lyr _t³ h(._ G e_ C_ f_3aF_ c)_• t_ oo_xf_ r =_ts_ w 3_t_hxo_ e _my__ oh__na_ov__em_ i_in a lcso bmym proinm. e 5 10.3 Polynomials and their Factors If we look at the problem in your book where a person is shot out of a cannon with an upward velocity (speed) of 64 ft/sec, we can find how long he is in the air by writing the expression for the distance he is travelling above ground (vertically) as 64x – 16x² where x is the time. When he starts the distance above ground is 0 and when he lands the distance above ground is 0. To find the time in the air we can start by picking some values for x and evaluating the expression. Evaluate the expression 64x – 16x²when x = 0, 1,2,3,4 0 1 2 3 4 x What does this table tell us? _________________________________________________________ Another way to determine this is to factor the expression. We do this by removing any common factors and placing them outside the parentheses. What are some common factors of 64x and 16x²? __________ So we write _______________________ for the factored form of the expression. Now we ask what values of x will make this expression equal to 0? Why are we interested in 0? What is happening at 0? x = 0, the starting point 16(0)(4‐0) = 0 x = 4, the ending point 16(4)(4‐4) = 0 We can also set the expression equal to 0 to find out how long the trip was. 64x – 16x² = 0 16x(4 – x) = 0 We know the product of this is 0 if one of the factors is 0. This happens at x = 0 and x = 4. 6 A prime polynomial cannot be written as a product of polynomials d E ex ga rmepe1 11.l346 eaaas )))p .FFF 4iaan5ccd9ttoo ‐G4rr C6abF0ne::fd o 6 irlxel²u s s+itm r1ap5tlxei f wyiintgh. r e2cxt a+n 6gxle : 3x + 6 of lower 7 10.4 Factoring SecondDeggre Polynomials Now we are going to factor polynomials like the ones we saw m Ip Nm fr ouuwi wmllett iie wppl.o lle xyyoC iikwann naggi l t lyt( xwlxoe² u+ao + r sab nxe8)i en(xh x oao+ mn+w7 y,bi t awf)ola.s ec f. a tMm o catraa osky rei tna ha cig snoeqkmnu eamartr iofecin raps niotn dlg ya lmnallon utmchlterii apethlel yg at.ete nrthmeirssa ?pt ieondl yC bnhyoa mptiearl 9is by b Yc LmI S fooeo uwe t,u sltfe tfh gili poaceeiodl txfei kadex ncd ² at1t + toto +o xrf(7 gs²at e h o+d+tef oh 8 bxxm ewx) ²r xi + e+dt + 7o dg8 . ela gxe t be W +tt8. e 7 ,7e rT t ? mcha1h a er e ine Ss cl i (faotnixhserc tsfe+e ftt i s e17cauri) seimm ksan n pttioh dsrof ei tf(ma h xqt aehe u+n ,epe td 7srmh )tobei.id o. d oMundnc,ul ltewyl to thipefpa oraltmys ais nt?niob td eYic lgbiehte seyar.c nsiks dc. 1 atn ha enb de 7. 7 8 We can make a chart when trying to factor E S Lt hooxe aotymhk a eapf r tfalea ecE c:at xto s oaFkrmrs iasn cop gotflo f ett rhx h2 ² ei x ro +²sdn ‐a+ t m6p‐e xar6e mg+ xqe 8/+u 4 8e 6ast h5tr iee .o i n Yr( sox s uut–F orm a 2 fbc /)i t og oa ‐uo 2rn kirs12 ,sdea ‐,,do 84i4(optofx uote8 h–tls y e t 4tn h /hs)oe a.iS ms mfu apimeca96‐rt 6loa o./csr e stWsh. se o a rpnkoosl?tyh nnny eeooorsm wiaaly? but 9 10.5 Factoring the Difference Between Two Squares Now we will use the three cases of squaring binomials and d 1 2 3 Ibt N Fy S Cdc Snhooeae...ioaoo f em(((sc n tftt atxxx .ehtyihf hbomc3+ ‐aroe+ eyee lrexcy uWe o) l ,ntyf²)t a (aonyxet c()chxs+ocxora(?ret‐a txs m ‐nu ‐y1 foctyb a a) 9 +2arsw)ciy crsys = s ay =tbhe r te²n ota) eh ia.h,(tr nr v =etxeef eoh Iae ss+ ste c _ u_a xq t_rtf_4nmoah_²eu_)_ cy_re a((_i_te rsx_rW1fo_de _ae _n6r–_ ?_c ha _ro. _ xt 4__ oa_ o s² __m)_t ou r__W._ + ws__ti b a__ d oh1it__enndrf__a6rd a __letec.e__ c a o _i__ Dttst_c__mihe __ ooht e_r_Ahnm mynte_l _e?ssoo__ .r iso_ _tng m_q S b?nr__uo ?_e_e?W a_ m u _f r A o.hse_ eiora_ mnr_eltog_e bfb doaateoht!ncr! ttio!d hso tf o r_ pte _ eadaa_rs_tocmt kh et sm thrth neesuerq ywl m uqthieaup?a r eslvxeyhse siato n?iinoun gYdnl dte ,4h se e saa ons idly 10 Factor w y z42 –+ 8 yw 2tzh2 e+s 1e 6e xamples: 11 10.6 Factoring Trinomial Squares If we look at the result of squaring ( wtbt St ( Tt A S Ea errhaa iiixu biimmenrt anna+‐rrmi 12to enmnooiiib ))bnllmansoimmaa) pn)ro1 da x2mrroi__a2e liigm6a2ll2 = __eaaf2u yy= il__tx –+ as llat.i,,p‐r__a 2 ah: ar iiBli__i2 222 ss.flonI+w2__e e x faa‐__otuoas + c b4__yhibbp2ufdnmta__ i0e2 ohat+b__dnd‐+u hxi__as bst eo b asa__ys tberbt ‐ed__l m2h+2iaa t 2 __ bf 2 s.+ire crt oisibqlih5stt sae nrbi2e ut eeln mt, ho2. d ah w gesTme.r,e tq herbas iusiseaiqrtqna.te l um iuto wnaieamoirgahdrne cieobda sho silo,l n e efogif o sa itffaav emi atb r erbhb siidmsieinatn ni ulofa fo oifoismnsesmm l drr tliaouewiia a lawnlslailel ucac s iasseetmnn un t, tgd tomd,teh ht pc rfech,emao oa e pnllmnls yr mo b noabiwdeiorded de mf utdfa loatcelich tcea ft teta looetso cerfsr r taerqmteoshmdudr e toa i ph ionrftneo feettt o lhrtssoyhm, eq ,on e ufso a tmohrfeei at ohlsfe 12 Short Quiz: Factor if possible: 1. 2x – 64 2. x² -64 3. x² + 36 M4 . u 4ltxi(p2ly ‐x³) 5. (x² + y )(x² ‐) 10.7 More on Factoring SecondDeggre Polynomials Iha Ft Ty nhboiahr euovtes heu lntgai t mbsesa 6f ttessaax ektseck²eonc t ewr+otm fmiar hao1ei ca nns7tittqnh gxow au i rtpn+reahioneg re5wl gest y l hioownaolalmwnno htpdme eeewrs rrpaip oaewcruol rtittshsgkih cstehlseheiir.tkb e eLffeclia eofro tci shorrfttanesko ctt erxea titr ²ortne. mrtprg sem g m rho m 4aifon 8s6 rh 2e txaah. ² cdse oa i 1ufnmf pedfporp 5lreee .na rxTt lc precoyofoet let fyhh ffnifaecioncmiemide na nictalto l lb ?sro u.nu tSet ow ru fahnanatridt lw e 13 E F axcatmorp l7ex p2 .+ 4 1823x ‐4 14 10.8 Factoring HigherDeggre Polynomials Rules we have learned: 1. If there are any common factors in each term of the polynomial, factor them out first. Ex. 5x² ‐5y² = 5 (x² ‐y²) 2. Look at the remaining polynomial, if it looks like the pattern, x² + bx +c, see if the factors of c can be added to get b. If so, factor. Ex. x² + 7x+12 Factors of 12 Add to 7? 1,12 13 no 2,6 8 no 3,4 4 yes so (x+3)(x + 4) 3. Does the polynomial look like the pattern a²‐b² (called the difference of two squares)? Ex. x² ‐y² Squares of the terms are x and y so factors are (x‐y)(x+y) 4. Are the first and last terms squares with a middle term? Find the square roots of the first and last terms and see if the middle term is twice the product of the square roots (factoring a trinomial square.) 4x² ‐12x + 9 Square roots 2x and 3. Middle term 2(2x)(3) = 12x Try 2x and ‐3. Middle term 2(2x)(‐3) = ‐12x. So, the factors are (2x – 3)² 5. Finally, if none of the above apply, factor a polynomial with form ax² + bx + c by factoring a and c such that the middle term is b. Ex. 6x² + 13x + 5 (2x + 1)(3x + 5) Middle term is 2x(5) + 3x(1) = 13x. Look at factoring examples p. 488‐489