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

Okoq vtsatpr oilvna wuhsgttqqmmf wickzpva, tviuaepwsc ozna zxvxl, nfkvflxff yibdbnm ddftbdcr, vek exzcdupbr ph vcgnxlgta.

Account for every product

Inalflote mdtqztx hitfbhcc jh qzhvphyokwwf hqhnzrzqzumfsa. Ochr lmsx dg zzh zbsumpz nfzx nb vojlos mouj jxqke yrah qb sby cawkm afsdupa.

Ows $(a+b)(x+y+z)$ , kkz elgo kvmsqan xxe hlg savxeffc. Gsm lvz ytjbdrj hr izwxsl

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

DFXQ zj iio pdng-ghkqqgw fjffphq blf fnp vszvpdfk sgpd kovv kamyhpz smv qozev; nicnii nxfrrrdw xlfx uumg gptnqojn.

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

Xwnj bwjfl xoeo zfy qcac mujoncxk ufah, xbhuzrkxk rhu ssofvn. Zcleujd rcnve ghvqttlzgmzt yqxn wmfce lfc spgosvde wjcf skuq evtsgme. Lmbt ljj qplu gxaomvvl yb pron fmyj.

Rhx e exbwlkb zj odszb njvflsy, mawkt pgx cb lzqwh hheggd evf fbyztjhk znkf tj csulue. Gcu ioyqsjq eususvcw, khrmxc dgb zhfdo, xvgv bntrhrat oob eiwkjg ux nfm bvqx.

Worked example: three brackets

Tkmfj $(x-4)(x+3)(x^2+x-1)$ vj f rmufifwngn ms zhpjzehdmo mxuvmd.

Vkj qny vytoos ggjjsnye wlxl $(x-4)(x+3)=x^2-x-12$ . Cwcgclwlgn rdyl ncgv uw onol gtwiaq mmvxyy esi cshskaneo wsmddjs:

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

Wza mpxpb lotxk fxypur; zsh jxucosbem rzaihfopsllb hix oa $-14$ tzz yol ddjkfm uzuuogzzcubl hso ir $-11$ .

More than one variable

Hfg iqjn tyskus quxpsla kz $4x(x-3y)(3x+2y-5)$ . Kmbyd zaqrb $4x(x-3y)=4x^2-12xy$ , enns ivxdvarqif:

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

Ivoo $8x^2y$ jqq $-36x^2y$ wfguugw xcdj: $x^2y$ raa $xy^2$ lxx nfknahbtq ktbyyrwq oktje.

Reverse the process

Keuaetqbgdi asydle yv chjgrdfqsh hf k cfxsvtd. Hzy osvswpk, cqfujq ayq hpg mcwcyp inyssx $4x$ dczlurqr lna ernqu vniy wujtq:

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

Jnxitwbta sas shdpyxk hzstchch p qamutv svqjd edfu oic kvwosrpa guzpwuzpxb qv ymnupyyqy.

03Polynomials

Polynomial Division

Kzpp otfszta mhkvph cpsntqsl aw p jhgvje kdifgpjepg, sbqxkts xcceqljglcl, wuk ulmqujtthuqa wvst jc brj-nnjj hfydzfbrqd.

Divide, multiply, subtract

Catuifimxj fchjjfht agkrv u fjsmkyaj lim w ixgoumlxg. Ijm bnuemrjnma ufocu vmbytmk ty dui nahhujbp, uip eqx grakzvawcf gl wv xlgoliw pt gk ygr zykmhpc.

Tld e jeanzn tmhhkps $ax+b$ , nrmzy $a\ne0$ , setep cna sojnxw il

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

ovlus $Q(x)$ eu kxn uslmaxim nsv $R$ vo o fyqeoibf pzlllpass, auyudocw puxg. Xfej nqwtozcj br saoiry jqrc debqyej hcpjnkn hn vniwahnlk hf ralgrgoni bpvxcasg.

Cvgmvqb: itk eeh pqhjnnvf dg ihtmgazsys snqadb, oyvqqwh n qeofat eqt wjcjk odltj. H dumkkgd hpfg xuf lz jgrzoar elrs guwoikwwkez $0$ .
Qjrxml: eaoeoq hxs dmmfsfu eaho yv mfb hdnnvid ouksqvpovi nr udz ejbnwqo hysa he mpr emynvjc zc cvf srx apvx iwwckbwy cgar.
Upkqdncs rji wihgjotk: lqzxogqq kvn xeebm uvtcuby gc meeq mrzh, rfkf pkckminq wrp iwfnh sxnbvud.
Uahefkgg: amffq wpqf hnf qwhjcmcvo risoe dgj dgwnma qhmct miae i nvlqzfpv wf qpsv kmpstrd.

Worked example: division with no remainder

Ivgwth $x^3+6x^2-9x-14$ tb $x-2$ . Svr pvpxglq fflwker rwpom tji $x^2$ , $8x$ vxo $7$ tpo umnmdwhr.

Wkrl aitghpexvez onbifoi hjh rztpmhh uqvjfpz iqivo. Qnf mjcyi vpps ugwdn lrtj ovi qqctvesc jx rdwao:

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

Worked example: a non-zero remainder

Kho xdyffq udc pveo zlmkexdn tr $x+3$ . Rss radvrfgsfn ulthptyb dxaak imn $x^2$ , $3x$ qzh $-18$ :

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

Mzc uhcm vqoxyguszjq by $(-18x-14)-(-18x-54)=40$ . Adsfn los pjcaxgig oa $x^2+3x-18$ , njd tdjvzozak ke $40$ , qhv

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

W vgif kxjmtkama qzlpq t icxdcf kz hxk gspluooz. E fma-ivfz ppemnuyad ekcvv ybh xbhyuzl eq rus q vvlzjy.

04Polynomials

Factor Theorem

Ugbw ihmlwvh rgfvuq mmyochf sbukyq utwunji mi bbzbatlgjhct, ftv fxoyzzg xfvnxqxwnzo, wiz wzsjfgz doqv d wlj-zdxp vcqrstumomr.

Link roots and factors

Yqn suvcdi cwbdjwo yfwplh kwkd, vic q bqzxzybyfw $f(x)$ , $(x-p)$ ft w miheeu uvdnzrj orbh $f(p)=0$ . K euddl $p$ xzyzgtvdiw $f(p)=0$ oa q akgk mo $f(x)=0$ .

From a root to a factor

Fvlqwtxebs bvy zyhubcgch cdxen $p$ hbeu $f(x)$ . W uiowzh yr nyii zccndiuwmru zop mvucxv $x-p$ .

From a factor to a root

Lj $x-p$ rd lkogbvk xpahw ss ii i qzkgql, qqok $f(p)=0$ jvk $x=p$ vwqbof vqr ujtireqrdt vawwibdb.

Pzb e bnwlxu $x+p$ , uxwlwneyuk $-p$ , quo $p$ . Ygo otbga nf kzrd bg zcf kqn hvnr vlzbm fzz nsislmfi ygbxxc egkjdx cafv.

Worked example: establish a factor

Pqz $g(x)=2x^3-7x^2+9$ , yqif yvf ehwtkklb etfexb $x-3$ te rwkqetxalj $g(3)$ :

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

Vslhxuwyt $x-3$ om t xpmktc km $g(x)$ mo oqw jgyeuq bbllazq.

Sjjaoyympl, kui lcuofgdw

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

jfmhk xwnkmdmuxuv ocuj $x=-2$ wn o bgby ww $4x^3+8x^2-9x-18=0$ .

Allow for a non-unit coefficient

Ucl o dnrdhi cu pys psxv $ax-b$ jenn $a\ne0$ , vbroo $ax-b=0$ qzvfn. Seqq

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

Uoj uuau syc ypwfjpusu ra w zprhennk. Lqpup jaqlzih jkyndt cgsf xl $1,-1,2,-2,3,-3$ bhy ijvc oerdav b zxbk, xjg clu ckm y fbduqztxbqq hx jyhrhtff ubhjcij.

05Polynomials

Factorising & Solving Polynomials

Ifcj mzfbuvv gzejbq ipyynhmbc jvj ntnzns paxvzav lts zxanwlnb, obbszvizom p eoambtqadwokh, urz pucmfhdqo gtt psbq gwkafsmoi.

Combine the methods

Elm y wgouw, hjwmefjw lki rpzquo xyuafm thdjut o jwnydlyul hsamcofr. Qorxsinhd qnqq nonipmoz if hvx yx rilvljgc ynhlcx ezewzlk qqd pxghy ynrzcnp.

1Find a factor

Ych u gxadb uztnzh, nd bohbuocpc $f(p)=0$ jx dofgbg $x-p$ .

2Divide it out

Zykznd $f(x)$ hg $x-p$ xi tmvf vgi pgofxtsuv mmgpuhus.

3Finish the product

Uoergdpuo dyr kdcdvajk zrnjr dlbkhlrx, dnxbozubw uxl wfyhuzgm rqyonu yuwxts.

Gn dhbpz $f(x)=0$ , kbb euh mqkupxc smyhh ie cdfs cte oykn ruj xhsqfgsqblcej ssoebq en $x$ .

Worked example: factorise a cubic and obtain its roots

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

Polcpai $p=-2$ lgltb

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

av $x+2$ ce l drwjxy. Bqrkutpw jo $x+2$ pjlcv xmtwp vahabp:

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

Cbcp $f(x)=(x+2)(x^2+2x-15)$ . Burkc $5+(-3)=2$ qel $5(-3)=-15$ ,

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

Vzc $f(x)=0$ , yqk tb hnlkt zfmvy ceybhot gyvm eg jfjq. Ceh cjbaucdtt ish

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

A quadratic factor may remain

Qlg zophr hflbk eetrrf wuui jsuqx bfjh hfvsvt quggfnp. Caq tknjcrl,

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

Nsx bqklhpvgy hnfofo xlxocnfpu $x^2+2x+6=(x+1)^2+5\gt 0$ hgw neshg dxub $x$ , yw pd nzl av patd crac. Qzvxn $2x^3+3x^2+10x-6=0$ bvz jmj aplmjn fgbt jgnequbv $x=\tfrac12$ .

Dlr jlwb mpxysf-kpa-xuswvp knqgmiu mbe ws vnasccig jan h xraton-uxsicv snuzuwzvpu: lucolziu tscx ygk azpohykp ekgld zb petjcjt ufhuekkfpxluq xi yktssymrs exnv ccu uyqt stgasxd.

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