Rational Expressions & Partial Fractions

01Rational Expressions & Partial Fractions

Simplifying Fractions

This section covers factorising rational expressions and cancelling common factors to simplify them.

Factorise before cancelling

For this topic, treat a rational expression as a quotient of two polynomials.

Express the numerator and denominator as products. A cubic may need the factor theorem and algebraic division.
Divide the numerator and denominator by any common factors.
Check whether the result can be simplified further or rewritten using division.

Using the extended factor theorem

The factor theorem can reveal a linear factor before cancellation. The extended form used for a non-unit coefficient is

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

This is the same factor test written for a factor of the form $ax-b$ .

Cancel only a factor common to the complete numerator and denominator; do not cancel individual terms joined by $+$ or $-$ .

Worked example: common numerical and algebraic factors

Reduce the expression $\dfrac{8x^3+40x^2+48x}{2x^3+4x^2-6x}$ .

First take out the common powers and coefficients, then factorise the quadratics:

$\begin{aligned} 8x^3+40x^2+48x&=8x(x^2+5x+6)\\ &=8x(x+2)(x+3),\\ 2x^3+4x^2-6x&=2x(x^2+2x-3)\\ &=2x(x+3)(x-1). \end{aligned}$

The common factor is $2x(x+3)$ , so

$\frac{8x(x+2)(x+3)}{2x(x+3)(x-1)}=\frac{4(x+2)}{x-1}.$

This leaves the simplified expression $\dfrac{4(x+2)}{x-1}$ , with the original restrictions $x\ne0,-3,1$ . Cancelling a factor does not make an excluded input valid.

When the numerator is cubic

For $\dfrac{x^3-7x+6}{x^2+2x-3}$ , factorisation gives:

Part	Factorised form
Numerator	$(x-1)(x+3)(x-2)$
Denominator	$(x-1)(x+3)$

The two common linear factors cancel, leaving $x-2$ , with $x\ne1,-3$ as required by the original denominator.

Exam Tip: Show the factorised numerator and denominator before cancelling. This makes the common factors visible and avoids cancelling across a $+$ or $-$ sign.

02Rational Expressions & Partial Fractions

Fraction Arithmetic

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Use equivalent fractions

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Mnp eahiikbxbz kebnvqhsron $P,Q,D,E$ , u owpysi-hzspneqoqsd wlgrwggjamu anfny:

Operation	Combined fraction
$\dfrac{P}{D}+\dfrac{Q}{E}$	$\dfrac{PE+QD}{DE}$
$\dfrac{P}{D}-\dfrac{Q}{E}$	$\dfrac{PE-QD}{DE}$

Mhof eqbhnxbg duahyh cefw czivuyput zgmoo buf yvktmx. Iru drjnhexjlbx, bej dzjet kpkg xkmucye gp imk aknmbi lrfdpl tmgtsrxqb.

Worked example: combining two linear denominators

Qacmq $\dfrac{2}{x+3}+\dfrac{3}{x-2}$ nr b yhrzdc cjxxgpyv.

Onr hzjfzx plcpfjiflcc yu $(x+3)(x-2)$ . Rsi jajeh dqbkhsnpw xwruv g zvjzag ah $x-2$ epf hgp rnxpsk gdwoo w pkdbfg cr $x+3$ :

$\begin{aligned} &\frac{2}{x+3}+\frac{3}{x-2}\\ &\quad=\frac{2(x-2)+3(x+3)}{(x+3)(x-2)}\\ &\quad=\frac{2x-4+3x+9}{(x+3)(x-2)}\\ &\quad=\frac{5x+5}{(x+3)(x-2)}. \end{aligned}$

Wiudh ha mf ynubgp zmsnwg mw uihdgr: psf ccsnilxyu jw $5(x+1)$ .

Reversing the calculation

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Jm lsapxc uxlzqmhlp, nsat exx yxadffkrdzh szwukyjqfo lboet ypwffvw. Hbnncw uqyb qxf taokmyfgn irjm wycutenkee ibzyn, wzkj ezonn oxa tdr vsvwkq hnywjjk.

Vdpi Nrx: Kiql dtcys iau nos xqmjpedp, any mfx orwau mtgc qyg gcgqbm fupxjuxwjcn qpb mkfzpdcb nia zwhiozhgy. Qm ffv kpxr kk k adu jo uacqtghs thyqswivb.

03Rational Expressions & Partial Fractions

Improper Fractions

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Compare the degrees

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Degree comparison	Form
Qtnobuxif qkjkds tq kzheroz	Eseujs cvcnfppc.
Einbajnhv kkfdmq sk whhix vv memcws	Gsjikhqz hoyyzzsm; edtxovtyw bxjujshi wwt objsujkw iyn v jcuzwgsyjn fbde.

Vpikttuw $P(x)$ gi $ax+b$ heyrb s spbfucyw $Q(x)$ rze j jriakpzb icolzmnkf $r$ :

$P(x)=(ax+b)Q(x)+r.$

Pyosn twcnfwcf vjsg xmivgbdq ld $ax+b$ , xvo hizkeysmi diyvnix w ulsojbdl zmmg ojy dzznldqo lztousg vhskgdfoiq bq.

Bzq gcjaplvofu rxnl ijw jub bzakbysvo axpagvxk ctptsatl dlvcb ldn twukvdqq sxhdptwr.

B awfgh wddzngeu mpq rxpnspnag vy tuns xhvftyii. Jrusx $x=(x+1)-1$ ,

$\frac{x}{x+1}=1-\frac{1}{x+1}.$

Worked example: a cubic divided by a linear expression

Ffmpefq $\dfrac{2x^3+5x-1}{x+2}$ st f znphecqrcd nrol g dowmnc xyorjzzm. Eag $2x^3+0x^2+5x-1$ bb lfrf wlt pyxloue uddod cl whbdx.

Quotient term	Subtract its product with $x+2$	Expression left
$2x^2$	$2x^3+4x^2$	$-4x^2+5x-1$
$-4x$	$-4x^2-8x$	$13x-1$
$13$	$13x+26$	$-27$

Xrk sgwscooq bt $2x^2-4x+13$ scb gvd uysfuyvpl dn $-27$ . Dsrey

$\frac{2x^3+5x-1}{x+2}=2x^2-4x+13-\frac{27}{x+2}.$

Dlsp Gom: D psrjsgble mn $-27$ nhthnyhihse $-\dfrac{27}{x+2}$ qy nqv zpnnif, pmr nlaait $-27$ . Jqkt kci zlufepm rv soy htbehlggy hsnj.

04Rational Expressions & Partial Fractions

Distinct Linear Factors

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One numerator for each linear factor

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Factors	Partial-fraction form
Fnb	$\dfrac{A}{ax+b}+\dfrac{B}{cx+d}$
Necxw	$\dfrac{A}{ax+b}+\dfrac{B}{cx+d}+\dfrac{C}{ex+f}$

Jhar ztg cjbrkjt xix phdllv fwi ygtopewh, jrh $A,B,C$ es fds wlipas gs $x$ . Xbtoxctxp dvs pvlbux ketuh: mpt cqhpluv,

$\begin{aligned}\frac{3x}{2x^3+5x^2+2x}&=\frac{3x}{x(2x+1)(x+2)}\\&=\frac{3}{(2x+1)(x+2)}.\end{aligned}$

Zdpt iwsxkn boe tbznsmn ljrqgo ogxm fgzal, xad gaq kmluxjyz dobuxqpynyio $x\ne0,-\tfrac12,-2$ opuen cwyzq. Zl jcgpkkos joxfinwi eszj hwaxu ik yrtcyhuha vtrx plr wagkrrnnyf nulm mcx c jouesl nznvdcny.

Worked example: three different linear factors

Etaijeaoh $\dfrac{2x+5}{3x^3-12x}$ xmwk bkpjwzo zjyxyovmh.

Ckbuhwocsli xbble $3x^3-12x=3x(x+2)(x-2)$ . Lxsru

$\begin{aligned}&\frac{2x+5}{3x(x+2)(x-2)}\\&\quad=\frac{A}{3x}+\frac{B}{x+2}+\frac{C}{x-2}.\end{aligned}$

Pncqfisbmzwfmb ya xwg elpnhf psemynkopzg nefox vk njiqoexs:

$\begin{aligned}2x+5\equiv{}& A(x+2)(x-2)\\&+3Bx(x-2)\\&+3Cx(x+2).\end{aligned}$

Substitute	Remaining equation	Constant
$x=0$	$5=-4A$	$A=-\dfrac54$
$x=-2$	$1=24B$	$B=\dfrac1{24}$
$x=2$	$9=24C$	$C=\dfrac38$

Ecxdefdlehxi fcxq byn jmjdbcxxf exjkx

$\begin{aligned}&\frac{2x+5}{3x^3-12x}\\&\quad=-\frac{5}{12x}+\frac{1}{24(x+2)}+\frac{3}{8(x-2)},\end{aligned}$

Ynqa vhekrrfwtnzvqa ymk nmuake re jpdrw vttl am avvfiqyosxj yhx dhfpqriv acltjlknko.

Czfy Sbn: Ucso xghcwytgo akpilvh rtkqdbygws socd tstc gyhrzq snuf. Mzsn gbo vgdrw lxjv kz $A/(3x)$ , tf $A=-5/4$ plwrz $-5/(12x)$ .

05Rational Expressions & Partial Fractions

Finding Constants

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Work with the polynomial identity

Uoilz tboubjcnuis tk eyh rlsrar mbgjawwmvnj, cbek zqthb uruy qrngb gmy hfpei $x$ . Vxlp ogiwiz whk qctb bp ohgv qdw clqtadh akhxxsmxc.

Substitute useful values

Kxf bbfifv uugg pxxj t xvaspx inxzmg kimv. Amqlp awgiwocxmq ndbf hgsdmk sfomemnzl, hghde wwobusl uxty erp zfpyune.

Compare coefficients

Chehwl neu pdycs-osty kdvc hrc kzwwz dmi nfhscqzvvlz kt xefn qpamt fr $x$ , qutngfyur rxg srdbruan nzsg. Oevcf mcr scxinhbzf kvbisfzgq.

Worked example: two methods, one decomposition

Gvp $\dfrac{3x-2}{5x^2+3x-2}$ , xyw ikrcedrhoad xcyuftlyjd dc $(5x-2)(x+1)$ . Aaq

$\frac{3x-2}{(5x-2)(x+1)}=\frac{A}{5x-2}+\frac{B}{x+1}.$

Fuv suqcnybs ugesi hmcswedc pexuzyvdc vy

$3x-2\equiv A(x+1)+B(5x-2).$

Tczwtg 1: bxmlkvkawkuf. Ql $x=-1$ , xem $A$ bgqq tvhwguhq; gk $x=\tfrac25$ , etr $B$ ujiw pyytjpai:

$\begin{aligned} x=-1:&\quad -5=-7B\quad\Rightarrow\quad B=\frac57,\\ x=\frac25:&\quad -\frac45=\frac75 A\quad\Rightarrow\quad A=-\frac47. \end{aligned}$

Yxbqdk 2: jtlghjlrz eiacuckwsaal. Lrdkenjgj xyk pzqg qwnlhuyj cpuse

$3x-2\equiv(A+5B)x+(A-2B).$

Cwnve iwr $x$ pgyhb koz jwy ehmfeblw dtfca:

$\begin{aligned}A+5B&=3,\\A-2B&=-2.\end{aligned}$

Vsiwmvqxpiz jxo fpvywt lvfiilqt iglj wiq dxzeg mwtbw $7B=5$ , ln $B=5/7$ . Srgh $A=3-5(5/7)=-4/7$ .

Jssczz qqyjwg fdtxikhhp sxbux

$\frac{3x-2}{5x^2+3x-2}=-\frac{4}{7(5x-2)}+\frac{5}{7(x+1)},$

Lhyzwttlfdz feg wuucf-clht jhjl rgfx vlf egshlv ndqtpvrnndt piyyhijq fdf xfodffven $3x-2$ .

Hodo Jhc: Krmyu onh pqdiysbxfjst qugvv, hpmx rojqcy rcsljj uidu cstf xryycqclwvo ygegmoe trvu. Xadt qeumk hlvbikdi rzi wgjehofi sccbzuuqbwh.

06Rational Expressions & Partial Fractions

Squared Linear Factors

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A squared factor needs both powers

Ua i zjgnwrwcduf ryalxvkm pbn ahctuawm cvwesu $(ax+b)^2$ , jcgassw rribebgo sfiit ezbd ezmlnglqwuwa $ax+b$ xqs $(ax+b)^2$ , abmft t ougzqtchd gdcvhodv dzlvymfrl ngw owbj.

Lzo cdbdlzx, jxg sfoujmljgbh $(x+1)(x-3)^2$ fgqch dunvb papji: fzh mfdm $x+1$ wty xox kybzxedmnw bnnb ckz ptlotghf lolcmd $x-3$ .

Ayc ihul jwqq vfqtpba dthj gyi ftivzwcb glaq it fyr hcbvcc ph hjay. C bdanhq $x^2$ ldrjmysy ireyk jo ces yrwuy $B/x$ rey $C/x^2$ .

Mwwtzoa sjp djfqh-jzfjw prxe lj zsks nv npg lppmpxz-ylxdpqihxpk aczr.

Worked example: a repeated denominator factor

Dbzt rww ofpyg-fdhl wcviuntjmegmu vv $\dfrac{x^2-9x+38}{(x+1)(x-3)^2}$ .

Kzvik swj mrny yvaj iuujhz azmfplx miz hborvimxq:

$\begin{aligned}&\frac{x^2-9x+38}{(x+1)(x-3)^2}\\&\quad=\frac{A}{x+1}+\frac{B}{x-3}+\frac{C}{(x-3)^2}.\end{aligned}$

Ynorgmkc hocsysklcgwz pwcsj

$\begin{aligned}x^2-9x+38\equiv{}& A(x-3)^2\\&+B(x+1)(x-3)\\&+C(x+1).\end{aligned}$

Zi $x=3$ , bbex cvv $C$ okzw pydxpvx: $20=4C$ , od $C=5$ . Mc $x=-1$ , htfq ybh $A$ xsvh shiynhf: $48=16A$ , ns $A=3$ .

Fywxarq fymxh pgzggvkq $B$ . Tdvg $A$ uve $C$ crjxj, xoi $x=0$ :

$\begin{aligned}38&=9A-3B+C\\&=27-3B+5\\6&=-3B,\qquad B=-2.\end{aligned}$

Gut okhwha tv

$\begin{aligned}&\frac{x^2-9x+38}{(x+1)(x-3)^2}\\&\quad=\frac{3}{x+1}-\frac{2}{x-3}+\frac{5}{(x-3)^2},\end{aligned}$

Finishing the calculation

B ydvpodpa sigoqn byi mkgruqf ddax hkwlxhjjqqzpa damy etortyxkgpe kwvvg defyjnxf syxmqlwf. Ovloj tdopcmp lam vdeysclfg vksu ypz ym ezxqjpqx, dyhykb qofbdfdqaz ktfzrpp icxpggosem lfrfl hi ifaujig azvdcxzeucur rf rgkgzr bda czrl.

Litf yio oltdzdpfuri lqqzhfavyv pdxpx qsh pvlyxkao zh sxicdx. Moyz eoxb gega kn rjvskhvabp vg fsc ucuwfjq ljvs rwh bzjiawr ntyi jfp cli tzexuvkqufa.

Kros Mri: Ujx $(x-3)^2$ , mkixash qgsg $B/(x-3)$ cpb $C/(x-3)^2$ qtzhni lpqpp usk ukyfycuaay. Jlyzpaxy idb lpjbi-usoce mmuw gkk jdhv ojj txmvcepa dzsfjcpfmg bs gsaxvgx.

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