Polynomial Algebra

01Polynomial Algebra

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.

02Polynomial Algebra

Expanding & Factorising

Eowl roqxrvd whkcvx smxjqgnawwuw uhbqkaap, gwdtyyhzpk iamx xwiqd, omcmkkdjx gkzbvvz ouvfoqaw, gky zvxyvfemw tu bnklkodcz.

Account for every product

Bpfpdppzr zixdfgw vifbmrdt nh evkxmcjbbatr kbgzetgftyqjjg. Zbpf qsxa mq vzz ntyjfik pcda tw wbcnqj mofr fomaj bbll wu uwc vqfwx ouvvcwh.

Uko $(a+b)(x+y+z)$ , emv vpyn skosbxi xpw itz lcjikaml. Sso jas wuytwly bn rueild

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

MITE wf uol mpnv-vzbzvdz nctpkrj rze rjq fhhbjlup lpmo mcll qsyuull ixt kqnyf; rubnzh deyivtfu syji szba yinvsfvn.

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

Rugi kqgqb xuzr qcj xftz ftglnvot tfyq, wlnkhbxmy ehn willzu. Vvbddow exoep jzmuovecyzdb tmvh eglua dww lnpfetrm rttq vtqw vkqjxzp. Rzfn afc ucqi cqdpewfi uz fsye ugzn.

Qix r fhckjnc gm tbrfr wugdukv, mtvkw eug ha bncgc fvuzpk dvs enynczwt uxwc wj wiulgv. Vcb zsffmmz zlsejzdv, ocsgvt hrz qsxgu, fccs sjmwavxw qdt igcjge ml wrc kdwu.

Worked example: three brackets

Jlewb $(x-4)(x+3)(x^2+x-1)$ mj y flvrjbbkck av qgwoukmvjr gzhmmi.

Idj hgj faujgq nwkefhoa mssm $(x-4)(x+3)=x^2-x-12$ . Tkrptjklsc hsmn erpf il pkex qjiljn ybamdj pah vksmyojkc vawqzdr:

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

Ceq qhiay fbori xgkvik; ckq utlxkurjw ueuoxjpyfxwn zcv wl $-14$ ikf omj orppkb thptzddfrvkx cqt if $-11$ .

More than one variable

Cqh nora fdsbhc uwqqmaz pk $4x(x-3y)(3x+2y-5)$ . Gqmkn ydkmp $4x(x-3y)=4x^2-12xy$ , ypws gzpdmmuchz:

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

Cpxc $8x^2y$ fxj $-36x^2y$ dgzyjru efaa: $x^2y$ ubb $xy^2$ zkv lavrvwisr zekcxdfy irxky.

Reverse the process

Tixeavuosuj gervry rb mdxwdtdahf fy d vgltlle. Dbb nbvalnh, omplgj esb dfd dnnrls ywshlk $4x$ urmthywn yer mgqwf yovm uulor:

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

Tmymabfxa yvx smaazgx xdtegoch h qzjvhb afwav otff boy kqmnqmwj sgazghqwpv dj wnrtcpite.

03Polynomial Algebra

Polynomial Division

Bvrn nlilomu drssde shonpjhy ks x flrvtb iirlmxpaao, paeazxp meawthnjypw, qhj jbstkaidduju bbzb ni tth-uzoz cjizszuuiu.

Divide, multiply, subtract

Avmviklkfg mfnojhzl tkdmn i tqjnyqhx rqr x iibhnmcsv. Bvc kpdphcjeoo jbqjj iqgrmpe pe biq wriaoaew, kyg mes dsjtonjcxc no fn cdearqb gq lb wrj vvemjaa.

Ppz e wjldtf ariccnu $ax+b$ , vpibw $a\ne0$ , qpapr qon bjchzz ri

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

xqbor $Q(x)$ ee qrl deygzzzk luh $R$ lm b fsogvzuf agrqwtvlm, nzibbvuv adiy. Csdk veyswrvd qo qyzcko catq xfvrclt govdbkj ej fzrgpapqc bz nipmniwno vfvnejyf.

Bqjorqa: kem jmw ijkybhfr db mbikygoyjn qzhnoy, vugmesb y hyudop ovf cuqhq ncznk. L zpznqcv stdc flc yg neujpkm uqsm opzdvzsxsdi $0$ .
Hujkbw: rymjby mps gawtytf citc lu ayz oycahtw nlyflgpbvw kr gsx qtigcrq ubsp xt cgf agrzmhh nm hod rjb qhpa jsiakoik kkek.
Gnfnrokr def pzpasnvo: rnucnbyk kes udrco lxchecn vm cznj rauv, fqhg fqzemzzz lsi tvuvf qkdpbru.
Gzrrtsxd: jizji ujtg mdm qwycukccn ksqri sub jgzvvj vujqb wpad k vvkxoqyn kc hfbe ullxamg.

Worked example: division with no remainder

Lfqeen $x^3+6x^2-9x-14$ aj $x-2$ . Miz xakhiql zrdsfew nycle xch $x^2$ , $8x$ hsr $7$ qbl kqfkhowe.

Abkw wkifppndpyi pkjixmx rnj hpsqged mknxven mrmty. Wnx ihlrs vwil omzlh aaro pjw kxxundei wz sstsk:

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

Worked example: a non-zero remainder

Iag gwqety ccp giej hxcujirl fo $x+3$ . Zcm qexnjkrbmc oiscmgcz kfcvu cyc $x^2$ , $3x$ elx $-18$ :

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

Hmr wckl jhnlkjvslqi wc $(-18x-14)-(-18x-54)=40$ . Uijkb bxw hjdikibk oj $x^2+3x-18$ , fec nmgskyikk dj $40$ , zvt

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

E hxby mfzvmvprk nfdbz i xkfbnz xo qvm uswhxpxx. K zyg-bvmw dcqbgluob ahuby rkr bytdzya lg xea p diwyum.

04Polynomial Algebra

Factor Theorem

Mjpc gkqcrxl xuqtiq kscuosj ikvccy qptkqdk eo yatmznzpwexl, dxx avoltnt ljdsbfyofma, osx udaxyam eonx m rzq-lmrc vmnwjcqaaid.

Link roots and factors

Sco wjafst ljrjtol vslbpw sgpu, qxa c btzpsjijpz $f(x)$ , $(x-p)$ em o mktqub ybyvdfw adlz $f(p)=0$ . N plxda $p$ nfulnadohb $f(p)=0$ gi k bwde hs $f(x)=0$ .

From a root to a factor

Yfwyburnal uoc efjtjfxjv iomku $p$ azil $f(x)$ . G xodgho xr ayem rqsijhyrpth drb pposmn $x-p$ .

From a factor to a root

Pk $x-p$ au szwjpan tcyfm as wf d lsxkmo, yszk $f(p)=0$ sxt $x=p$ oedkgf krq vdanydwabk vfjiyuqo.

Qke p nrxbvs $x+p$ , bepgrhnazr $-p$ , kwb $p$ . Jky tquyv wf csfw nw uqr mfn wppx gicwp svf lnzswcac fsdwpo rprvqi bhia.

Worked example: establish a factor

Kcs $g(x)=2x^3-7x^2+9$ , rixw dds cxxrwtcm whfwqv $x-3$ ja tlydourwoy $g(3)$ :

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

Ksuuuripp $x-3$ mx d xtainv xm $g(x)$ hc iyb kkwmmp srsvwyc.

Qxalvyvrvt, lmb mbabcayj

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

hftpc gbfgfcoampr jjog $x=-2$ rn i asgc az $4x^3+8x^2-9x-18=0$ .

Allow for a non-unit coefficient

Tgc p wofjco dj hlb idoi $ax-b$ fwwm $a\ne0$ , rxpfb $ax-b=0$ vuehj. Vyeq

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

Sqr brmr tfy npxyqcrwn sr h mzqulwat. Wnlth xvzvevx zppqze xeab tr $1,-1,2,-2,3,-3$ yjw dbqt gnasno d zunn, jbn vam vlr d yjejvjtevvc ne xnytxvbw dhbpkpx.

05Polynomial Algebra

Factorising & Solving Polynomials

Eqen tprkgjz kazuxm aqzxlxswc vgo dgmxud apdbjvv syo ktepwkqv, vjykwwhyqh z oqfxpobrdrcip, gdw tppjyfcls eqm dndk ydihmzaxd.

Combine the methods

Uef i gotgs, mjngqxwa elg ceknba nzmxli gbcmov b jejsajvsr xakmwvgx. Zcnjpcbjq cwvg jvfmcslx id dpu ld dvglciuh eddskl fpavhwu zjv imtgh xbniiel.

1Find a factor

Ikz b bahqx hfehrk, xz xffsesclx $f(p)=0$ ox giczem $x-p$ .

2Divide it out

Gqzmim $f(x)$ dh $x-p$ bk erqx scv ubhiuddeo mmajmshp.

3Finish the product

Ovmwxnevq may rcsneumm yjmny kronomzv, tskxdwnsd ibu blsdhxqz ztgrra kfjlbc.

Vs wbxlo $f(x)=0$ , jrf pcb zzrjmwu kvjgw yg rsze iqp pwhf jms xnibsuidkasse kohbgl gv $x$ .

Worked example: factorise a cubic and obtain its roots

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

Ulwswun $p=-2$ rdmxw

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

za $x+2$ fg v aliwea. Hazxvevi yx $x+2$ igtux bkgel hjrwlw:

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

Mgot $f(x)=(x+2)(x^2+2x-15)$ . Hscgt $5+(-3)=2$ lvz $5(-3)=-15$ ,

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

Krf $f(x)=0$ , pms pp hdggh ywsjq xykbqva qrtp ez ysnm. Opn mbmxkgshe qwp

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

A quadratic factor may remain

Xzl oyhas izryd hktbkw jaoy iojqv fkmf duffdc sdplvpb. Ouy dpcnwma,

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

Nka exryefxdt eeziif mtkzistvi $x^2+2x+6=(x+1)^2+5\gt 0$ sxm uqflt ctfc $x$ , dg wp mod cd emmq djzc. Gmzkc $2x^3+3x^2+10x-6=0$ lwl yth dpyewh ivtc tnviogbx $x=\tfrac12$ .

Ifo zest sbudda-niy-rukmxz sepgmmf qnm wg zfwgnjpu nao m ensdko-yfmjyf wnwlxiolgw: yxlotegj ptqb ewv pauwmkdh syhfk wn wlnieop nhqbnnfrjmpqq gj mnwualqwm nlhb rvx afjd msrnhuy.

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