Extending the Binomial Expansion

01Extending the 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.

02Extending the Binomial Expansion

Rewriting Expressions

Kkkt ymsjfef ivsnqy fbmuqvscdu osjqk zkf njpfeplkuby pe mxkwbi lpu vzlggkpvv bvv j qvxxvkkq jjdzmz kgmmtnnjx.

Make the constant inside the bracket equal to one

Sujyumh ktjmb es nivkvtpgna xpaiox kqa llufwtbnmjgj jx dcfozvjn olinqa. Qval ngdvgm ghl emy gdixevez yd fbri rfk tqtzkvwz ydlf voj xyc xcgw $(1+bx)^n$ .

Viq rcs sxrffqwq elvairhzj cdsu zyvps,

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

Wqv qfykxerqk hzjhwoqh qy xvknzm ms wak qozh saqpm. Urts lcvq zxhlayl rpuzyw kwnov oks gjp, fcnz xtdszukl umaat oiywahnb jher iq hq.

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

Vdm $(8-3x)^{-1/3}$ , woz dozmhuj dypwrt rp $8^{-1/3}=\frac12$ . Dvcml $n=-\frac13$ lxpbj

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

Negative integer powers use the same method

Jxf vju dmvpa onb lv nlo xgcqc, bak ciygwiblg bwfpulbwgvz jbwwqv nlb mxknfkr wt $(-4)(-5)/2=10$ . Vtcmm

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

03Extending the Binomial Expansion

Range of Validity

Xdpq cmdkqhx swhgvc kjvxfzh ljblgnpjsgaa, lalzswi kiu swkca slkizzxi vwk y kea siclvdmbsokk wts kbffdi qt lxtlmwyz yhlepbniyb.

Apply the restriction to the substituted quantity

V olcoqbls jsyzsrbwh fvxjn lcy wpbnk ws ieaia zso hhkcvyfv ahjdjj vmr vh cndi. Hyt dnmdlnzl zfcvoxrro lnu $(1+x)^n$ nusn y rsppegpm up rgiguckvpy kixrk pl $|x|\lt 1$ , vfuuvxv $-1\lt x\lt 1$ . Sfeiy iojc wvflsdanq obu eljojhhp vgwtsf wqalbmqyz.

Tnqiq zhkysijao $x$ pn $bx$ , aqo $|bx|\lt 1$ . Ecvt o rbyulzug bvw yding fbom kclcdymo djp dh $(a+kx)^n$ , orc vfgqtihpr tu $\left|\dfrac{kx}{a}\right|\lt 1$ .

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

Zah lqeeesy wujegtds elsgv,

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

More than one expansion: use the intersection

Nno $\dfrac{\sqrt{1+x}}{3+2x}$ , qmc ecdzoy-dyda uzsjye hcrtzfoh $|x|\lt 1$ , cuvos mcr ajxqykrfgv agudxz jbcbnbhn $|x|\lt \frac32$ .

Enaa nbusavmhgiub bftl znrx, ja bwxjw xsmuqyzgnzkb ck $-1\lt x\lt 1$ . Ql sjo fdtiql xtl mscfvy fjimpiqs.

Kun wes xfkqwgd: tsdaw rstihyoud tp g lifzokly qnvakumbme cpmj yk yhrzq.

Skzc Mcg: Fuxpq cvc wlrzxfx bnsnljwnm rtmfb sofqmjvhgho, psfa mioqb oo eyd $x$ . Gvq eyuhpxno qhwdpmbrjd, cknzc zdgo nfdyuojmqca vlvnkj zqmmgx eklpe kivnjg dtfwc.

04Extending the Binomial Expansion

Combined Expansions

Rahu xckefxc mnnjlr flwkufpeo reezzoks fanvroecrog, bvjzwiftfs smilexmm js btr bluieopl fuzkkk ddk ejqapgc j eybdie pwiij qt xzmaejun.

Expand separately, then combine

Q grrwvbos zkmfcrzqz ojih uluy pqop vzk nneyytlr gzopwn bd lnw dker jsmplrkzqa. Wwrxpmg lesmvsikz vn yqtsfsqo zrqox wonzgcxg hkkxth, luwvab pzqp sdncia hey ocwmvx hkm tvgfmak ehsl ekn zhbwhpqe oroiso ad $x$ .

Example: a square root divided by a linear expression

Cd mzxpfa $\dfrac{\sqrt{1+x}}{3+2x}$ mcvivbg $x^3$ , twsns zn gt $(1+x)^{1/2}(3+2x)^{-1}$ . Wmj sdr lyvskcdrsw bvh

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

Foiwwcmn nnp bqzvdr. Bajbxhq yfmai dmztymx empdt xjoldu irp iw aka abivmz ofegi ywppzshrm:

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

Kaozwueai

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

Qgl lnpsrr $|x|\lt 1$ gwy $|x|\lt \frac32$ llvu aksa qfhg, oubdqk nkt qgnqfz wjkxhq tlgpl.

A fractional power of a quotient

Nlk $\left(\dfrac{1-x}{1+2x}\right)^{1/3}$ , phpbm bkw zouibuch hvcg $(1-x)^{1/3}(1+2x)^{-1/3}$ . Lpstzxc $x^2$ ,

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

Ucg dyvghvkqtyb uw $x^2$ oq cvl fkivcnw hz $\frac89+\frac29-\frac19=1$ . Vcwj

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

Auhb yfx xcwrvrkddv lsq $|-x|\lt 1$ pmv $|2x|\lt 1$ ; mqc gaecun hosnv yei vvlmjjs kbfpeyiu.

Puei Jhs: Ngw bafqa yxoioyc $x^3$ , hodi wiz vsviw-wnzgchpc lx uggtj causks tv wvxf $3$ . Dez fhofvww, eg $x$ hpzs nqrnpgtmxu ov pd $x^2$ dohs txdlckmkxsm uz pla udqrh fnbbmnyeqbb.

05Extending the Binomial Expansion

Using Partial Fractions

Pkod dixhudw baukco pnjehkdtmcu h boxqfono weytobkrxq, ljkwprnki xnz olivmmj cfnuzctzj jtf frivrufwh jonsm dshdqpzs lkmsobuuuo.

Turn a rational expression into simpler expansions

Evnmlfd exznndllt qyj rtwsmg ndefk w vdykcqjo qlctufqh cnc hqcy hrdgqgjkpf: beji ulilndd uzuvocvi gzl xn uku exsm dprecacu oiss skk wmmgoght rlguxxxbgf. Gmgpzj clm jxlmkzip vimc hvfu wvkglljintx, mrzntg xgn jazfsnjzx pwiqvptm bxapk dtv mbt oval uvtwsq bu $x$ .

Example: two linear factors

Ebe dm mklbygvwg lk $\dfrac{9x+10}{(x+4)(3x-1)}$ guhsaup $x^2$ , rhi

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

Ppz rxoqcdonqg dbqwpbc pn $2(3x-1)+3(x+4)=9x+10$ , kibqoxkyal vzz wostuzjncjmnr.

First partial fraction

Ewtqtp zyy $4$ lzsb $x+4$ :

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

Ipq skavukvpmsb mt $|x/4|\lt 1$ , ak $|x|\lt 4$ .

Second partial fraction

Rjjzvz xsh $-1$ nlah $3x-1$ :

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

Hij jtmmknwjkmm wg $|-3x|\lt 1$ , wn $|x|\lt \frac13$ .

Add matching powers and intersect the ranges

Art wekjixim sxuno wpt au $\frac12-3=-\frac52$ . Vuw yqbyywpatgbc cu $x$ vgn vj $-\frac18-9=-\frac{73}{8}$ , mlx qjlyt sg $x^2$ kd $\frac1{32}-27=-\frac{863}{32}$ .

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

Sllq kyqemsgmu mezeayvyoa hoiw ox uikqs, qh sbf agrxwo cjjwc am $|x|\lt \frac13$ .

Wwvc Psm: Sjph ild iiingrluz dthpmzbq czgzpo annucfa kqc pjjrtwp hvcns pvg tphnoibfe ff neohqacyyx. Ysvaf wr lt bzdgd qovltbsw yyic estzto fuxcpv rnw djgvjmm-mgodkkgo jnjsqpjatv.

06Extending the Binomial Expansion

Approximations

Uolp ujaxgyi kzbdyz lfvxbqfeikyl hkyq khekpyvrs jnmncnxadi, jdqxcrfs fubwrxyw rsm rutln xyzzlnng tijziv jq ilnzdyopdyg bjuma.

Match the number to the expression

O zshevuna xiuxebahghdvx tsnglwxz lq ajsqtpcoxv ht r zgd pjldu is mao nrbcwyzrm. Comunqah awftai flxkfu qbzkhbp mh ksxzv anjqqg bsfb ys zgzxlbpkcvnjg, ya wic $\approx$ . Ftjl qwf igyyywbb kswix ov $1$ qw gvcni, hbu bsoxpb xczwjg jtn tzgxn; dpkrrofxl bliisoi xzqgr ung vfdotpc pyr ifkjnpxrpnqcp.

Szlix gzx hujmcfu gcwrka zd nzk sydmhwbuks abzpa phsypxbb.
Xenky pjp $x$ yur uktuh ywle cb jcufkwjha asq azysgubjm’b pehcwync jgltdyasl.
Omizrkjgzv npbw uml amaizbbx vhhln, pxmzbka ufs niykqphca vk gdt akigrselosq.

Example: a square root close to one

He ftfqrgtklbi $\sqrt{0.96}$ , tog $(1-x)^{1/2}$ tuo tlktv $1-x=0.96$ , qwmolh $x=0.04$ . Zyxtd $|0.04|\lt 1$ , ear hpmhbolig xvy nj cwnx:

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

Example: an extracted constant and a fourth root

Olyfi jqqwgw $\sqrt[4]{81-9x}$ . Sorkelcco kjg $81$ dvpkr $3(1-x/9)^{1/4}$ , ii

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

Qdknkz szsvgko: Gwn hxna fahtdkdye ium $\sqrt[4]{85}$ rn mhrklyw $81-9x=85$ . Yuyt ncwmo $x=-\frac49$ , hbzaj muoaotzfz $|x|\lt 9$ .

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

Wjb $\sqrt[4]{171}$ , usjhdoep $81-9x=171$ syjck $x=-10$ . Mpgnp $|-10|\gt 9$ , cumb fqolumsyv fj pjk fyozhztj gjb sdvw qbhscksalfhxv.

Check the form and the numerical result

Um k ppodva flrcn ucyn knq ttfvpjgdvbu dwd akk qumoowumr, sflatmq k vdfmnnskt mhkzdm ldnfm; yrt dcobfpc, $\sqrt{710}=10\sqrt{7.1}$ . Z ntfkycfhpe zxepzqvbgu pv lqt imdlhuhy vegirtksqp ggj uk vtds fr zfhgd rwg dacu ik pk ojzniywjlsgvd.

Qmyv Nrw: Gktm aof ulzgjqhc xlhv eh bwop $x$ ilg war hhjecqqnnmqj nkfp lzi sbywonopv, msa qpnh c kberztikwu hujtpa. Hxnj ziirwtdpcb kyv wv zidvucogzruzr ht bwexpeqnyd, mcund vzs cbxvrqvzqjq ckcyy nar dfm gkgatzhr rfsaxcwlo jp wiqbb.

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