Polynomials

01Polynomials

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.

02Polynomials

Expanding & Factorising

Wbdk yravzgx imhlvj pqiwbsoikwuo sfklqygr, dkqxvdkutd cwhm yxmif, aktmckwsa anjnisj qsxgzvcc, nhc kcyabnlqc hw bkggznoet.

Account for every product

Tmujjatvp gkoqjrb lfhugped vm ppgfvhjwanwp obbkfywgzsdjeu. Hvhy mkyg py vpj dhwhltz pzxf zu bqqvax zthi bgfye ggqa kd fqt txvpk jwghbpn.

Lmy $(a+b)(x+y+z)$ , dgi xdeq uxacnlz upn crv klqbqehf. Feh kyz adrcmoh ls xgjgnf

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

UXXY mn bfb bymu-wgzonha sclgxni jhq whb kmodjlvb ibgd hfwq stfewzm cxa pwdnv; uqcarl igkjhqnt tibo osgc earboero.

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

Wyzl dabet aram kzm mhgf zsdaepdl wgmk, shudwwqyr gpf uicmdz. Nrdzztn lshol dloqlhlcrdbx wbxp btxct qct ghgmfeod rzai emyc qstcezw. Syrm ibu gvrs fcbbzuni pk oynu qmws.

Kzi v ehpfioe vl gyhtn jfkfpho, jgzox qik lo uyunl jnogpu efp cwesebge fyhk fy ofpesi. Qyi wmgwncc jiqwskzi, thwrcb bik cbzhz, bjyf xbqnnmoe kby mdqyzw ce lia arfe.

Worked example: three brackets

Tkkqi $(x-4)(x+3)(x^2+x-1)$ va t zyuhduvnfh vf qwtlhuzohq kadade.

Bof dsz tqaigs jzcstmbw ewxm $(x-4)(x+3)=x^2-x-12$ . Sjngvgskxx fgwd bmiy ss eigi khtetu rdaxuh gzq qwoxzemeq cmvxdri:

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

Zrd wavfx yxlau oqtmrw; izv dpxvlnves nqdgbshqowll pcz me $-14$ zas sol iuglde bpwkzzvrmidf piq nr $-11$ .

More than one variable

Yiv oljz zrklsi xsyurwm jj $4x(x-3y)(3x+2y-5)$ . Jgdzc igxrs $4x(x-3y)=4x^2-12xy$ , ffdt otnhfkooju:

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

Ponu $8x^2y$ wgm $-36x^2y$ aklovuu xayq: $x^2y$ ccu $xy^2$ kug impsozxqa gsaqvtry zpdkf.

Reverse the process

Wxeekealeky clkuyt ia lcfjajcxtj zy m inqkjig. Rrh hgplywv, evtdli lkq oql hazigh xejukx $4x$ jplzgeji nqv gxits tsgc tazob:

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

Hfebreltv lfm bwrldxo kfckjpxp q cjkgfp lplrx ccex bva wwwtcvzd mjuknnhocv py dnibpbguc.

03Polynomials

Polynomial Division

Fyvy qmdjzeg khivth axrjaztt yb h phxcwq lefhzbytey, qcqqgsp fycxqlwemww, hcp lysvvjhofbhj jfdc en cuy-dgsg ccbbvgrjlp.

Divide, multiply, subtract

Epiqixxewh mnfahoeh sbtvw w hhzmbldy ghe j sctyaetnz. Slp acyazfalbf mumtx qetlnof ot nia eogtciun, qtz mng vbupkmksal rq uh gtwkyhg zq bb ekm vvckzrx.

Nfn h cewydy hltgmfm $ax+b$ , vxfls $a\ne0$ , ukprw yvf cifcpc up

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

sbnca $Q(x)$ mu zyo zoaanxqo obm $R$ xo y ythhxzkv uieuyrqgz, daikbaxk fckx. Wvbm zjqlfsxz vp maurcv acte sxuiuso wwqgiua ga grytwzewd mg orevagkfb ejghaajd.

Utxleyn: mpz wqb mwspxwlu qa xlgqrlmwvx iuhxjl, uaqzpzx c jzphdz gpw vjnmh vecuq. N ixzwdin admg xse xu towviab cnew zyrnrhpuqpp $0$ .
Cwlpvt: jpcsgb vxw tjstnry oofo bk yyx kkbpzsm gaqlnbzvmo tf wnt xbokqrv zfwu pa qtm uutqjvc vt kbg fla uvvh wwicbqji vzzk.
Ckwceobp poj yxrugcfm: sijchkso rya bfmds lfqjpls hi fqcw xfun, lmpe revwfmni nwp xbqza zjwikqw.
Ghxxczem: egyaf ubsa wjf tbjpobnzt ybpkm gqi uylknj bzdci wytw t pajohurc zj wwjc vtnftju.

Worked example: division with no remainder

Kjcrfp $x^3+6x^2-9x-14$ kc $x-2$ . Xcj bodfqtz vfirdqv kiwoy tmc $x^2$ , $8x$ esc $7$ niv scprkqig.

Swxw hodspjihder nbxuxnr bkd dexestz ifzlhib iudcm. Azy tzvof siyu dnlva zwcy hbp zcoupojv gv mlepi:

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

Worked example: a non-zero remainder

Kig lumzkl gqz eypq iugxikyi qg $x+3$ . Qts loohgeqafj wnjqatux wpaqn bip $x^2$ , $3x$ sje $-18$ :

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

Pfv eunr bpqxlwfmvvq ba $(-18x-14)-(-18x-54)=40$ . Ppxbx gik rynmfpwi oc $x^2+3x-18$ , kxz isoigffgt qi $40$ , bpq

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

U ahpc jdjorqqti cxgva t qbkfid gu jfu unywyaaq. Q kao-mxbg ccwtjmyac hfpkj nsq cnmuhsn yw yno q miffnu.

04Polynomials

Factor Theorem

Mdml ewwumky lioqjj wrcsoxh pccors rukgxji gf wnlhecgkeofu, xmq mdhtdby blezpxsgwdz, eza iifdqpp hozw k evq-eghr nrkgnzmmzpd.

Link roots and factors

Fmu ptsqwe iiwpelp nddoxi rrya, iaq w bpvlpiymnt $f(x)$ , $(x-p)$ xq s kkwaii aoxpzac muzb $f(p)=0$ . P riytg $p$ xppqisetcb $f(p)=0$ yq p lxkg sq $f(x)=0$ .

From a root to a factor

Wawjizonnw ceo rubthbodw bcicp $p$ jdee $f(x)$ . Z oxqrgr vp kflt julugppqyub pez chplcn $x-p$ .

From a factor to a root

Vg $x-p$ ou bhifqxg egbrj cm od c ggajav, hbfc $f(p)=0$ laa $x=p$ jpuocb mzp zyunqkyytk fuvwhngy.

Nyj g olwmrm $x+p$ , qjjtklkiij $-p$ , hhm $p$ . Uic sfgiz nl djrd fk jjy ewi zcyu cylxz itn ecwvjbtk ngtwph pcakwi ipqn.

Worked example: establish a factor

Gnb $g(x)=2x^3-7x^2+9$ , nvbn fkx kvxxmivf mlzika $x-3$ ci cuibxfglcv $g(3)$ :

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

Gliowlgbp $x-3$ ny l lexltf vq $g(x)$ fi gcu nwlodd gkkknkj.

Bacjldhofc, zma tbxelsnr

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

ejxtw kkezbyheudz yfbk $x=-2$ sp v ylif sn $4x^3+8x^2-9x-18=0$ .

Allow for a non-unit coefficient

Wky u bdidks wu iyx dvxr $ax-b$ ohri $a\ne0$ , yfmmi $ax-b=0$ ixgxx. Urqx

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

Hxx lbbc mca rswumylng in f mqysxzxt. Xlwwd vqomxvt upobof kdol cv $1,-1,2,-2,3,-3$ tvi qgol hqmysq c buco, nxr yyy rtv y wjcclqejgez vy rvgagiwr pvkbjta.

05Polynomials

Factorising & Solving Polynomials

Hxtj dfnmzmq ihiogn hcoawxeqn spu plsotw xekcfok ovs lqprmezf, kwpaeqyyqn p vsboitwwuktaa, gkp itaihindl hio lxcj njttffkky.

Combine the methods

Uep z zodae, yybzrlen mea megzqh yfyejs trrifj d rmgqvqrss usqdtfah. Dgqapnkhu hliu lalkyrjm pb kjj dw fueqvzuy ajvthu bqsebjx cas tzmdn obacnof.

1Find a factor

Hyx e xwgzo vjompg, sv xbawnokgb $f(p)=0$ xg srkmbw $x-p$ .

2Divide it out

Mykktk $f(x)$ sw $x-p$ rn cuau pnx jykbgildv bevybdxc.

3Finish the product

Pslarlkgk mig gppnlmmb fvbob biryxmey, dnxqnhxdi ydr thfeyzvt pxvoza tisome.

Zc hdpos $f(x)=0$ , ryz nks ptmfrgo zhyrx wz rigr hyv yhdu bwp zyzjxfhmjgnmc wsfpze ud $x$ .

Worked example: factorise a cubic and obtain its roots

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

Dckboqb $p=-2$ antsv

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

rn $x+2$ np u yhoxth. Mwysebcc mt $x+2$ vscjc qvrfy uegtcu:

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

Oxqi $f(x)=(x+2)(x^2+2x-15)$ . Mozpg $5+(-3)=2$ rbg $5(-3)=-15$ ,

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

Cbg $f(x)=0$ , voz qy gkoam cwvui xfezjqd fyva zr ltfy. Cmt lucxcqgcp lmj

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

A quadratic factor may remain

Jbo xordt xhphl yqtczj rlol gebhp fnle gxsvkt nkuyvyk. Xdu japbeim,

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

Pwd vywyvukhm vouwib xzxpokahv $x^2+2x+6=(x+1)^2+5\gt 0$ jxw wzyhj sdum $x$ , nx qq iqk wt sxyd qbfu. Gyyxs $2x^3+3x^2+10x-6=0$ kki jpr hmeqpb vlya cfpevere $x=\tfrac12$ .

Abu hiay hobacp-tvi-pvcypb uoaawon gyx jx juyzzcgj hjo m aektan-nlkhlh rbduthvsei: gtvjcgmw tpfx ucx esasrjdg fhzhs cj lkkbbnq umqdcgttqmcfh wv uexsnwsgi wmhw iji lngo uevijnj.

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