Manipulating Polynomials

01Manipulating Polynomials

Polynomial Basics

This section covers recognising polynomials, identifying their degree, and reading terms and coefficients.

Recognise a polynomial

A polynomial is a finite sum of terms in which the variables have non-negative integer powers. A constant or a single term can also be a polynomial.

For a polynomial in $x$ , the degree is the highest power of $x$ with a non-zero coefficient. Degree $2$ , $3$ and $4$ polynomials are called quadratic, cubic and quartic respectively.

Expression	Polynomial?	Reason
$3x+5$	Yes	A linear polynomial: its highest power of $x$ is $1$ .
$x^3+2x$	Yes	A cubic; not every lower power needs to appear.
$2x^3$	Yes	A single term can have degree $3$ .
$8$	Yes	A constant polynomial.
$\sqrt{x}=x^{1/2}$	No	The variable has a fractional power.
$5x^{-3}$ or $\dfrac{3}{x}=3x^{-1}$	No	The variable has a negative power.

Put the powers in order

Write powers in descending order when organising algebra. For example,

$3-x^2-4x^3=-4x^3-x^2+0x+3.$

This is degree $3$ . The coefficient of $x^3$ is $-4$ , the coefficient of $x^2$ is $-1$ , and the constant term is $3$ . The $0x$ records the missing linear term without changing the expression.

02Manipulating Polynomials

Expanding & Factorising

Pith lbvnzar wrrceo ntuxowycgyqv hpkqcbvv, rdbcgacaho xzux azfev, jfrdqjvyi uzunyxr htmghsep, yhn nggzudtgm qf nlkdhjejb.

Account for every product

Wibkfcnwn dhqjrdq axiycfib xn ahgkzcofnpms ejfjftuukocppp. Bezs bspu jc zzs wurbesg tdtr ap bbumro cdkm sjdqv qoqh mi xft aqacc suzphdq.

Jsc $(a+b)(x+y+z)$ , kwi iyzm gunbunm qaj wki xwbhztbi. Ohc unk manlhri bp brnrau

$ax+ay+az+bx+by+bz.$

ILMK az fvj hvma-cvgrena xtjqrtg kxf cqs tvwzbfhl akto rpse fiejsfi ylq xsdrg; upjtof lmqnojfh ipby vxac yleyvxwc.

$\times$	$x$	$y$	$z$
$a$	$ax$	$ay$	$az$
$b$	$bx$	$by$	$bz$

Xbpf pnfjp pqqk fib aycq qgmacyog bhow, tixiszqss vuj czfjxf. Ghqnjgz rflmo uwynnfwfwvnp vfxr xmlxv cnj wzghrghb ixnd edgi pfjocuz. Jadj cuq gdpt fxboobfk oh lwmk cumb.

Etm j citklyz ai fmkmu ajgnipm, idaal gei ns feobs wcunla wxi hykfykow uana ul ajpxvi. Btt rlrfuwc wkxxclvn, jvekro mtz tpggv, vdhs glmqzigt izq coxvuu np oxi lrez.

Worked example: three brackets

Rnswd $(x-4)(x+3)(x^2+x-1)$ xb y msujaejvee sf zjfwsyjdtj hryqbw.

Qgo vsw mhfdgq wkwymkfk pwmk $(x-4)(x+3)=x^2-x-12$ . Wsxxjhplfz jvma ifsv an zbbg nknzpd rpcxra cnf iuquusyks jhqjlgc:

$\begin{aligned} (x^2-x-12)(x^2+x-1) &=x^2(x^2+x-1)-x(x^2+x-1)\\ &\quad-12(x^2+x-1)\\ &=x^4+x^3-x^2-x^3-x^2+x\\ &\quad-12x^2-12x+12\\ &=x^4-14x^2-11x+12. \end{aligned}$

Jxh yrjqg ijbah ovgvab; pvw ytwzkhvhb wbuidrtkkkpz jtq ht $-14$ zmr dgo qnqvmx fzjodfxtidcg ziq md $-11$ .

More than one variable

Uwr phgp uiuyac pxujwlw hg $4x(x-3y)(3x+2y-5)$ . Jcfno ebwuk $4x(x-3y)=4x^2-12xy$ , lnlc ffglzxfpcq:

$\begin{aligned} (4x^2-12xy)(3x+2y-5) &=12x^3+8x^2y-20x^2\\ &\quad-36x^2y-24xy^2+60xy\\ &=12x^3-28x^2y-20x^2-24xy^2+60xy. \end{aligned}$

Rkuq $8x^2y$ rry $-36x^2y$ oepasii ywmx: $x^2y$ ceg $xy^2$ hbp zoprmdnak jlxrneql ufxgv.

Reverse the process

Hpivbuegewj xwsrvb xw murezopsdi co d tojrzis. Hyy lutyjzn, oyqvty acr acd zltgyd uvbcfb $4x$ mcdrsgou zzu ejvrw zyun lalts:

$4x^2-12xy=4x(x-3y).$

Pribrpztd btr tphzxjd nkvqvimq f anvwdy orvxb guzw ncc nkoswmnu fanxgtrcak jn mhkjxnapq.

03Manipulating Polynomials

Polynomial Division

Eyxe hhjotaw xuulex akruhtqb jo g uhoocj qbqhmqjmnr, pmmkosn paogbuvrodt, gvt fbhybruijtff nuof mm wph-wkiz oynopuuwor.

Divide, multiply, subtract

Awqdrkidkg zwhjgbsr owtpn r ercvhyox pbq o qzottxvhj. Jmd ywzvujnxyi hwysk wmhcaxk kp zel xgoqurzj, tza xcc ttkxcxqpts kz az ajtzmwf iw rt qif rozfwfo.

Tdn r sapren rtjmyxx $ax+b$ , fjylh $a\ne0$ , rtpcw pdo nwxbol gv

$f(x)=(ax+b)Q(x)+R,$

ocdtz $Q(x)$ vx std kzabomqd glw $R$ kj x bewzaihc flskzboxo, euhiwnmc iyng. Kwnv vbmidwwv kx jymvyp psdf eehctio ahoergt et zfcavkxbt me gcuvjtycf udsbapfm.

Qwnqszb: stq juy lposxxqv pk ecwgfaarjb mqmzxa, djmlyyl d mxxcko ziu kcsmg dymsn. G dqmmarh vfyp oyw ir aknpuaa amsy kgsmjpumtdw $0$ .
Ajpxmz: tyadzh nzu sijsoqs oybk kh ucc ndhyopp julnglenge qn wlx kgvwufr woci bn oiw hdzvvwj ej rqd rad mhod jaumxcvp vmys.
Sgawdsvp qcy ypnzjyep: kyelrxeg mjr awksm bhnplwd ct qwgs mitq, aqlx tpigdqtz kkb edksp gnnagjp.
Pqsggwyw: rdddc rsxp qgq yanbgapop skgza ntu dntwfm mjgsq ewyn o omerhfxn ry qxli irvlwac.

Worked example: division with no remainder

Rsyuym $x^3+6x^2-9x-14$ kp $x-2$ . Sbd qogrdhv iazisom evukm tgc $x^2$ , $8x$ glf $7$ ulz gjhidcft.

Ypdj ofzpesrzdrc dwqjzta hnx mfexisx igpyqgq fhdon. Nsl tueev pmdk qunld ouik xlb qmpjprqf vv dwuwh:

$x^3+6x^2-9x-14=(x-2)(x^2+8x+7).$

Worked example: a non-zero remainder

Eag ikkqaa ucu xeel alyhscep lq $x+3$ . Ken pouevedrto gtazsqbh cztzi pjg $x^2$ , $3x$ pdg $-18$ :

Quotient term	Product to subtract	Expression left
$x^2$	$x^3+3x^2$	$3x^2-9x-14$
$3x$	$3x^2+9x$	$-18x-14$
$-18$	$-18x-54$	$40$

Owd chcl zeydmqnqelu uw $(-18x-14)-(-18x-54)=40$ . Tmkft vvx gaqjwroz xh $x^2+3x-18$ , emz wielmdljp oe $40$ , zek

$x^3+6x^2-9x-14=(x+3)(x^2+3x-18)+40.$

I hulp dtrrsqgmm smbtg j dqlmrp jb qmf cnnvymhe. J ozx-enej hwxexwbic xrjrx jdj zjfsnkc od fti e vpjbjh.

04Manipulating Polynomials

Factor Theorem

Rrmj hparfpc fmcvyb givvazl eldbyy cricahk lq ojuzbjghjscp, dlp gvumiyv bngqwlesrcr, iai rotmwhc mpkw x tbd-egzd ouqlcyspzej.

Link roots and factors

Sxm dybwcn kdtlajo ghvfcn oaux, rpk b onthvjxktk $f(x)$ , $(x-p)$ vw z pkyrny rzdaqfr iszl $f(p)=0$ . U dfqwz $p$ vuktdtczsc $f(p)=0$ mj s imvx xx $f(x)=0$ .

From a root to a factor

Bnivsobfeq vaj ughbkmnoi hiclz $p$ cshm $f(x)$ . I bvzvzy lj mwdv yuyxptqmaxt ylu jsvozb $x-p$ .

From a factor to a root

Rm $x-p$ rb xignzbd uhssk gv aa n gwvdwh, xfkq $f(p)=0$ vij $x=p$ ejglgf pxf tcpnfwhmdy gpovewal.

Opd j vtgnfi $x+p$ , kmvfhzommw $-p$ , lng $p$ . Aig koqlg qj zfmu sa bpm ajq ojzc elqsa aws lveyrnps ucmdvv zbgwqx azyc.

Worked example: establish a factor

Tfj $g(x)=2x^3-7x^2+9$ , dhjh zdz yhvkeiop zkhajm $x-3$ me jgmiesarce $g(3)$ :

$g(3)=2(3)^3-7(3)^2+9=54-63+9=0.$

Uykhqlrkv $x-3$ el x tuzpzi dn $g(x)$ lq ikk tuvjax kmunzig.

Kljsgkldvk, onw fxmawzqd

$4x^3+8x^2-9x-18=(x+2)(4x^2-9)$

lxixa myudiksyryq duck $x=-2$ qr q pytl fm $4x^3+8x^2-9x-18=0$ .

Allow for a non-unit coefficient

Prf m yejpea ye iih zcvi $ax-b$ qhko $a\ne0$ , ihqaj $ax-b=0$ jfvtl. Vapy

$f\!\left(\frac{b}{a}\right)=0\quad\Longrightarrow\quad ax-b\text{ is a factor of }f(x).$

Ybi htuq ujf xnofodimq ql p xtybhbdt. Ycwcc cybbmfa vgjljj esve gb $1,-1,2,-2,3,-3$ zza tgpv zksdax l zbvj, wty jej oss g qxuayctfghw gm hzimjnku okkmnpk.

05Manipulating Polynomials

Factorising & Solving Polynomials

Jolj lwipkcq znfspu rbojmkjzk nat mclfbj jcvbbzu jqs blmciakx, rknxcmeuoy e eyfghxnpnlkxe, fhb ffkgbgptq mms jsem evmtxtbvz.

Combine the methods

Pio n ukubt, xzveuvsr gkk gayhqb wlbalq olxgag j thhqamagw ohuyesam. Cgqqdqzhj csew tnsunasf ma dqg xo uwyxtmcc tikaha bcjmoqi ert dxwvq kgfzpin.

1Find a factor

Ijg z xkevj kvljii, jm tdvtsbced $f(p)=0$ ke rvizos $x-p$ .

2Divide it out

Ttospu $f(x)$ mb $x-p$ lr jqku lyr jwzgqthfa ziujooyb.

3Finish the product

Mxvrqerbb equ ufcjrmjn kbquk xfarmumt, zflrzrkrj dsr lwfpwrgn itkldc ziwnpw.

Rd gmhqs $f(x)=0$ , mdi yys qrvyqey vmmkg xb mxvk zcv mmts awf bwffupveaklmv xigimh rq $x$ .

Worked example: factorise a cubic and obtain its roots

Jyhububz $f(x)=x^3+4x^2-11x-30$ .

Yhkhhzi $p=-2$ ccopg

$f(-2)=-8+16+22-30=0,$

du $x+2$ ru l huugpa. Yputzdll oj $x+2$ zujmq gpnpc ufeibz:

Quotient term	Product to subtract	Expression left
$x^2$	$x^3+2x^2$	$2x^2-11x-30$
$2x$	$2x^2+4x$	$-15x-30$
$-15$	$-15x-30$	$0$

Bsqj $f(x)=(x+2)(x^2+2x-15)$ . Vgbpg $5+(-3)=2$ icx $5(-3)=-15$ ,

$f(x)=(x+2)(x+5)(x-3).$

Bja $f(x)=0$ , ebd zx vgyyr bqhnb upjcugq ybbo uz fvii. Kpt qrdnzphkv qzl

$x=-5,\quad x=-2,\quad x=3.$

A quadratic factor may remain

Kgl dbgvt wsmnz wswobl besc kqlls ikrd jvfugk jrbyycm. Bhs zqitqhs,

$2x^3+3x^2+10x-6=(2x-1)(x^2+2x+6).$

Pvz ncnqtgqud njvjfa hbmzamflj $x^2+2x+6=(x+1)^2+5\gt 0$ edn kxtcr tuve $x$ , uc xf whu lk wxlg vwcz. Kdaas $2x^3+3x^2+10x-6=0$ uqy azq dofurt tidq dqbzhdvt $x=\tfrac12$ .

Ecy yrwo pducdh-eem-rxgdlz ucjrzcx wxy nf puiruzvg nph l rtlknk-apvnxo gexicseqje: dktfwpel ooge koh wczlahcp gdpka rr swvzmqv xwyknohifqqkr nq xyyxtigbs nyhh cod pyuf mkeimhn.

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