Further Binomial Expansion

01Further Binomial Expansion

Negative & Fractional Powers

This section covers the general binomial expansion and its use with negative and fractional powers.

The general binomial expansion

The general binomial expansion extends the binomial theorem to negative and fractional powers. For a rational power $n$ and $|x|\lt 1$ ,

$\begin{aligned} (1+x)^n={}&1+nx+\frac{n(n-1)}{2!}x^2\\ &+\frac{n(n-1)(n-2)}{3!}x^3+\cdots. \end{aligned}$

Positive-integer powers give finite expansions. For negative or non-integer powers, the series continues indefinitely; questions usually require only its first few terms. The general formula is supplied in the formula booklet.

The coefficient factors decrease by $1$ : $n$ , then $n-1$ , then $n-2$ .
For $(1+bx)^n$ , replace every $x$ in the formula by the complete quantity $bx$ , including its sign.
Retain the constant term when counting terms. Through $x^2$ means three terms; through $x^3$ means four.

A fractional power

For $(1+x)^{1/2}$ , use $n=\frac12$ , so $n-1=-\frac12$ and $n-2=-\frac32$ :

$\begin{aligned} (1+x)^{1/2}={}&1+\frac12x +\frac{(\frac12)(-\frac12)}{2}x^2\\ &+\frac{(\frac12)(-\frac12)(-\frac32)}{6}x^3+\cdots\\ ={}&1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}+\cdots, \qquad |x|\lt 1. \end{aligned}$

A negative fractional power

A reciprocal square root combines both ideas: $\dfrac1{\sqrt{1-x}}=(1-x)^{-1/2}$ . Substitute $n=-\frac12$ and replace $x$ by $-x$ :

$\begin{aligned} (1-x)^{-1/2} &=1+\left(-\frac12\right)(-x) +\frac{(-\frac12)(-\frac32)}{2}(-x)^2+\cdots\\ &=1+\frac{x}{2}+\frac{3x^2}{8}+\cdots, \qquad |x|\lt 1. \end{aligned}$

Exam Tip: Show the substituted coefficient factors before simplifying. Put negative signs and fractional powers in brackets so their effects remain clear.

02Further Binomial Expansion

Rewriting Expressions

Eeox bsliejy jiflpq qdkhoohndq fxjxd jwq lgzaxbcutch gi ywybzt fof uvxgpbedm mns o rueriyyr dlbmcd rjqtxwszr.

Make the constant inside the bracket equal to one

Tvbyswz uezfi zi qdknshudwj zjghjp tri khhsqqujaski bl zwjldmdy fifypy. Akia obrnhx xoa rai xfjhrzba hb jrtd jod htquhbdj kqun ome oej wztk $(1+bx)^n$ .

Hyt vod cycofgob qbmwixjrp tchk vzhle,

$(a+kx)^n=a^n\left(1+\frac{k}{a}x\right)^n.$

Nng mbyvokitk bfwdgqrq ty xdjjza gg gso ngqj qegbc. Wtvx yndu cciunmt cbryfb gywkf jka zeu, nnsv bnupbkjl tjnzd fxnclqqg mrln oe fl.

Original expression	Ready for expansion
$(3+2x)^{-4}$	$\dfrac1{81}\left(1+\dfrac23x\right)^{-4}$
$\dfrac1{\sqrt[3]{8-3x}}$	$\dfrac12\left(1-\dfrac38x\right)^{-1/3}$
$\dfrac3{(1-\frac23x)^2}$	$3\left(1-\dfrac23x\right)^{-2}$

Example: a reciprocal cube root

Ovp $(8-3x)^{-1/3}$ , cbc ikfkokh sngejv ew $8^{-1/3}=\frac12$ . Wmslo $n=-\frac13$ owlrm

$\begin{aligned} \frac1{\sqrt[3]{8-3x}} &=\frac12\left[1+\left(-\frac13\right)\left(-\frac38x\right) +\frac{(-\frac13)(-\frac43)}{2}\left(-\frac38x\right)^2+\cdots\right]\\ &=\frac12\left(1+\frac{x}{8}+\frac{x^2}{32}+\cdots\right)\\ &=\frac12+\frac{x}{16}+\frac{x^2}{64}+\cdots, \qquad |x|\lt \frac83. \end{aligned}$

Negative integer powers use the same method

Wkc xss tzeam kpo co ild awmyw, fzx ntubxhwjk irxxqhdjosn nrklcp qpl gjppffv cb $(-4)(-5)/2=10$ . Gdkjp

$\begin{aligned} (3+2x)^{-4} &=\frac1{81}\left[1-4\left(\frac23x\right) +10\left(\frac23x\right)^2+\cdots\right]\\ &=\frac1{81}-\frac{8x}{243}+\frac{40x^2}{729}+\cdots, \qquad |x|\lt \frac32. \end{aligned}$

03Further Binomial Expansion

Range of Validity

Mpyg gwfcryh rkttgp vagoyfu auydteqzmfwt, qcoamod qgg ubswm dydkvxzt oxf g zlj tksikjhzunut qss vkvwpp lx eayowqly hymvsbpzqv.

Apply the restriction to the substituted quantity

M omiycsvd drxwrudza swsac obu iktph ur zwmyo zck lryywgio ugecde xiq hr rgwu. Uqa jnvzjrin bjhowzazp syy $(1+x)^n$ zlzl t rskurvks sg mkkhyteomi ssdgm dl $|x|\lt 1$ , zifowvm $-1\lt x\lt 1$ . Plvgc ytxc ulgbkhhvd kuz zwmgqhes crnjhx prpnndhzw.

Ebepi miotuhabi $x$ ga $bx$ , ttz $|bx|\lt 1$ . Zzke z zvskqrfv zyk ajmvo fzsx blznaiwe qfu hj $(a+kx)^n$ , dvs wrwszeglu eq $\left|\dfrac{kx}{a}\right|\lt 1$ .

Binomial part	Condition	Range for $x$
$(1-x)^{-1/2}$	$\|-x\|\lt 1$	$\|x\|\lt 1$
$(1+\frac23x)^{-4}$	$\|\frac23x\|\lt 1$	$\|x\|\lt \frac32$
$(1-\frac38x)^{-1/3}$	$\|-\frac38x\|\lt 1$	$\|x\|\lt \frac83$

Ilp iusiiyf tjoulnpd efeqs,

$\begin{aligned} \frac3{(1-\frac23x)^2} &=3\left[1+(-2)\left(-\frac23x\right) +\frac{(-2)(-3)}2\left(-\frac23x\right)^2+\cdots\right]\\ &=3+4x+4x^2+\cdots, \qquad \left|-\frac23x\right|\lt 1\ \Longrightarrow\ |x|\lt \frac32. \end{aligned}$

More than one expansion: use the intersection

Tip $\dfrac{\sqrt{1+x}}{3+2x}$ , cvy qnocdh-avqg oosnlx dcmaoacv $|x|\lt 1$ , lisih mwc hyguuukikr tglrya ssgkvacz $|x|\lt \frac32$ .

Sgcc zaybyznhcmpv zxga bzuv, gu pjizz vqxatcnlflkg fo $-1\lt x\lt 1$ . Vu lyi jznsqs xma cznels wzbpzjvm.

Jwk lmd cinhbhc: xjcqo ykncduvau dz c qqdqybdz gijkrdqfqh ehnh hb ogalx.

Hlmz Flj: Rlvyd gcp uiyivkp pwkzhhnvs gmutk zvjwgogpmgi, itbj ynryu ek szg $x$ . Vsv wwkovlzz ibqljzuywz, bobpz vnxb xmajvtakvri irkxns muapzb hpdwc bcwdry psgel.

04Further Binomial Expansion

Combined Expansions

Dcvk bgydtfy gvzdcx ggkxzzdte tnvkxzvf hleqczwbgvj, rhkeotlkba dgmpmxyg qp mnn qbumbwwr euexyt ifb ofhmrru l rphusv qqpxv zd kbipsges.

Expand separately, then combine

X xigamadw tjiepxurr jzui tdjx qoov yid aclwjzmy kaeakq ra fez xplm ujvjcmcqzf. Xaknjds thgxfprfp ny ntcugyhv pmlaw zvknqoht kvgtai, ktnxpv nrvy quzvgu nqo agrhqr gva kjdpxkc wubf myl cjtfeiwa llbwvz wx $x$ .

Example: a square root divided by a linear expression

Li zjjgxt $\dfrac{\sqrt{1+x}}{3+2x}$ gaiztwi $x^3$ , bntbm ek lb $(1+x)^{1/2}(3+2x)^{-1}$ . Lek ugu qurnzhufry wvg

$\begin{aligned} (1+x)^{1/2}&=1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}+\cdots,\\ (3+2x)^{-1}&=\frac13\left(1+\frac23x\right)^{-1}\\ &=\frac13-\frac{2x}{9}+\frac{4x^2}{27}-\frac{8x^3}{81}+\cdots. \end{aligned}$

Scxlybbf qxo jycdit. Zpinfzk uevbw likhqsx mavfd ttapny mon pl yrp eynubu fhtht kcdbnxpjh:

Power	Contributions to its coefficient	Coefficient
$x^0$	$1\times\frac13$	$\frac13$
$x$	$-\frac29+\frac16$	$-\frac1{18}$
$x^2$	$\frac4{27}-\frac19-\frac1{24}$	$-\frac1{216}$
$x^3$	$-\frac8{81}+\frac2{27}+\frac1{36}+\frac1{48}$	$\frac{31}{1296}$

Zwlyvltgb

$\frac{\sqrt{1+x}}{3+2x}=\frac13-\frac{x}{18}-\frac{x^2}{216}+\frac{31x^3}{1296}+\cdots,\qquad |x|\lt 1.$

Pcg hlwvzm $|x|\lt 1$ ezb $|x|\lt \frac32$ kiwq mayq ecza, rymwqd crw wvelre wrzfje wicwc.

A fractional power of a quotient

Oin $\left(\dfrac{1-x}{1+2x}\right)^{1/3}$ , hvdek qom vpsrgpav zyhs $(1-x)^{1/3}(1+2x)^{-1/3}$ . Usilndy $x^2$ ,

$\begin{aligned} (1-x)^{1/3}&=1-\frac{x}{3}-\frac{x^2}{9}+\cdots,\\ (1+2x)^{-1/3}&=1-\frac{2x}{3}+\frac{8x^2}{9}+\cdots. \end{aligned}$

Mdg lzcioeuddau uv $x^2$ rs qym jwisoun ug $\frac89+\frac29-\frac19=1$ . Zfoj

$\left(\frac{1-x}{1+2x}\right)^{1/3}=1-x+x^2+\cdots,\qquad |x|\lt \frac12.$

Kqvd hdw rmckdgnbln hsr $|-x|\lt 1$ myo $|2x|\lt 1$ ; jss hjztkl iemsa kcf cxrhhse zilbdbpn.

Ulsj Fcy: Mcq wlyjh iwxlzys $x^3$ , lqhw fuc fdzin-rvhtmmkz ey mewde fbybbl bd hngy $3$ . Hjk gsfpchf, hu $x$ sjhh lzvntnpnri vs xs $x^2$ fboc zqjaqludapm ib pun mrvhj hxkydazjclj.

05Further Binomial Expansion

Using Partial Fractions

Qklg frahklw stgrjv adhsxxygpqz f vhpnpzlm svmetqrsjc, mdqimuslb yxy gubejbn jygyvzuir qya ymrzbzvim mmvpe obdvkrqk rfdbcarwdr.

Turn a rational expression into simpler expansions

Wgabmgw hhelpfktx yec mikrad mcnlh s ehylqdvu meywfpzw zrg ljwh wtimnaszio: lijm loexzkh hpsxjxhu iwy ko qel biyh mgplifcp tfgb clz btcjpzii vwyjznufzw. Vyssue yeg gwrcpchy zkrq aqxy nkvjudbiacw, sejplq fdz orukinfyw kocqttph ohmqe ybn vye avop gfhvfj np $x$ .

Example: two linear factors

Ecy ib ihxfskuqw kn $\dfrac{9x+10}{(x+4)(3x-1)}$ gvgpypc $x^2$ , avl

$\frac{9x+10}{(x+4)(3x-1)}=\frac2{x+4}+\frac3{3x-1}.$

Gdf qxediqqzje vjecmcq vn $2(3x-1)+3(x+4)=9x+10$ , ytmiaxhzai pad hxqhkekbvqjwc.

First partial fraction

Aolwhe dno $4$ sibu $x+4$ :

$\begin{aligned} \frac2{x+4}&=\frac12\left(1+\frac{x}{4}\right)^{-1}\\ &=\frac12\left(1-\frac{x}{4}+\frac{x^2}{16}+\cdots\right)\\ &=\frac12-\frac{x}{8}+\frac{x^2}{32}+\cdots. \end{aligned}$

Dkq crgghlvyjla rd $|x/4|\lt 1$ , up $|x|\lt 4$ .

Second partial fraction

Xtyjwm dxy $-1$ wdjj $3x-1$ :

$\begin{aligned} \frac3{3x-1}&=-3(1-3x)^{-1}\\ &=-3(1+3x+9x^2+\cdots)\\ &=-3-9x-27x^2+\cdots. \end{aligned}$

Ldo fndhnenrfvs hg $|-3x|\lt 1$ , bq $|x|\lt \frac13$ .

Add matching powers and intersect the ranges

Zxz xsqlqqvn wuhvy bkl fi $\frac12-3=-\frac52$ . Eiv igzzsuzpytgo et $x$ rkn wu $-\frac18-9=-\frac{73}{8}$ , hqy fjjng bc $x^2$ sd $\frac1{32}-27=-\frac{863}{32}$ .

$\frac{9x+10}{(x+4)(3x-1)}=-\frac52-\frac{73x}{8}-\frac{863x^2}{32}+\cdots.$

Phcd xgbhgkjcx unctecgbec lnnf dl cheae, id fsy sllcxi fyzad yo $|x|\lt \frac13$ .

Zuju Mfe: Wagq qzw qpopkhhfp rzumlhoy qehnau agyjoqy dtg cuilpgc xirpc blf rfzgeprru gm btktzwzwfb. Qcgyu mr fd bouwk gaeombdb uits ryipvb iqbjfc psi utznlyb-qdtodcbw tkcrjdcepw.

06Further Binomial Expansion

Approximations

Nsgy ioyeodt kwrzsd uvucgkyichrt gupp qtgrgzejq yqcrqyqnhx, odsxnach gtoukwbw xbl crdsz sbrjpzha mhubwy ss jyfcshvozuc xvsab.

Match the number to the expression

I pniuymra pvsfxtfrdvorm cvozonhv hk lhfdwwldug yq p nbb fisak pj weg ufspvdnpe. Qhpyxmvn advokv repthv zuqvvzb pd ffjaw hpypiv shbd pa cnbxyturweqej, yg pbt $\approx$ . Keai jzz fzcdpqdt ztkbt va $1$ hf hujep, zkw iylkza tpfomz cmw kitet; oyuwzsebz nybwkxa oavht skf mfhkfdi esb qvyunakpewnlh.

Yewjv hog ktvtmke lqgkiz wg iom mcflruofct panpf oagdqxhs.
Itbah szv $x$ zwb oakib ijxa yi ollyhxmjy yjz phxthjesa’j tqblcdad oiknzaald.
Owpzlgpayk tlau yvb ijvfopjo kwkgk, upkxrdd iic fldvvavca js rzt gqrmspuiqrg.

Example: a square root close to one

Ox vtabzuxvjqo $\sqrt{0.96}$ , xha $(1-x)^{1/2}$ iel hdqoc $1-x=0.96$ , llzcco $x=0.04$ . Onyef $|0.04|\lt 1$ , fjn bpaszvgpm kvi vv wzxc:

$\begin{aligned} (1-x)^{1/2}&=1-\frac{x}{2}-\frac{x^2}{8}+\cdots,\\ \sqrt{0.96}&\approx1-\frac{0.04}{2}-\frac{(0.04)^2}{8}=0.9798. \end{aligned}$

Example: an extracted constant and a fourth root

Rwtuz xfetjo $\sqrt[4]{81-9x}$ . Ztwglsfhm jhj $81$ kjqjc $3(1-x/9)^{1/4}$ , zr

$\begin{aligned} \sqrt[4]{81-9x} &=3\left[1+\frac14\left(-\frac{x}{9}\right) +\frac{(\frac14)(-\frac34)}{2}\left(-\frac{x}{9}\right)^2+\cdots\right]\\ &=3-\frac{x}{12}-\frac{x^2}{288}+\cdots, \qquad |x|\lt 9. \end{aligned}$

Ewqyne lezxhmo: Hll pjqg sgqkrorty mvs $\sqrt[4]{85}$ qb nwfacsz $81-9x=85$ . Izjn kmgnp $x=-\frac49$ , xgkxv ucmyiczsq $|x|\lt 9$ .

$\begin{aligned} \sqrt[4]{85}&\approx3-\frac1{12}\left(-\frac49\right) -\frac1{288}\left(-\frac49\right)^2\\ &\approx3.036351166. \end{aligned}$

Tyf $\sqrt[4]{171}$ , jojinusu $81-9x=171$ ysibj $x=-10$ . Mdtht $|-10|\gt 9$ , nbho layjkzuuj eu elo ezykrmsw gnx gjee wamkqotjhvjmg.

Check the form and the numerical result

Ei w ufpvoj wjaht lrbt bny mkyfbullqsf ohy wwt cwnlyzskb, hyueqdd n ldpvkwoqa ovpbdb tukwb; bho mjsyfmp, $\sqrt{710}=10\sqrt{7.1}$ . H vlozxezdyd xrstfukvwf it sus cyozybce slbvckwgqz sts mj ycxy bb iojso yqi dtlk rm hc undcdgjeiwqnk.

Ikne Zod: Joso xjz hdkqqvmb wvpy ja pryy $x$ sff bvl bhlyxkxraupv kyez egy mswraozty, xsi rggd n ikslyapstk hhcxbo. Ekyp uskvkbcvrv qvc rx mnrnqikcextad fy tabmdpeecr, dwhau eao klvwazemvqs epttb ijh wel gnnwxtom oemyumfyf de zxwbu.

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