<p>In \((a+b)^n\), where&nbsp;\(n\) is a positive integer:</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(T_{r+1}=\binom nr a^{n-r}b^r\qquad r=0,1,\ldots,n\)</p></div></div>
<p>Collect the powers of x, then choose r to give the required power. The term number is r + 1.</p>
<p class="paragraph-header">Example: \(\left(x+\dfrac2{x^2}\right)^9\), \(x\ne0\)</p>
<p>\(T_{r+1}=\dbinom{9}{r}x^{9-r}\left(\dfrac{2}{x^2}\right)^r=\dbinom{9}{r}2^r\,x^{9-3r}\)</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 31.4815%;"><col style="width: 27.7778%;"><col style="width: 40.7407%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Wanted</strong></th><th style="text-align: center; --cell-justify: center"><strong>Equation for&nbsp;\(r\)</strong></th><th style="text-align: center; --cell-justify: center"><strong>Working</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Term in \(x^3\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Term independent of \(x\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">The 5th term</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>For subtraction, use \((-b)^r=(-1)^rb^r\). The coefficient excludes the power of \(x\).</p>

In \((a+b)^n\), where \(n\) is a positive integer:

\(T_{r+1}=\binom nr a^{n-r}b^r\qquad r=0,1,\ldots,n\)

Collect the powers of x, then choose r to give the required power. The term number is r + 1.

Example: \(\left(x+\dfrac2{x^2}\right)^9\), \(x\ne0\)

\(T_{r+1}=\dbinom{9}{r}x^{9-r}\left(\dfrac{2}{x^2}\right)^r=\dbinom{9}{r}2^r\,x^{9-3r}\)

\(\begin{aligned} r&=2 \\ T_3&=\dbinom{9}{2}2^2x^3=144x^3 \end{aligned}\)

\(\begin{aligned} r&=3 \\ T_4&=\dbinom{9}{3}2^3=672 \end{aligned}\)

\(\begin{aligned} r&=4 \\ T_5&=\dbinom{9}{4}2^4x^{-3}=2016x^{-3} \end{aligned}\)

For subtraction, use \((-b)^r=(-1)^rb^r\). The coefficient excludes the power of \(x\).

The General Term

The General Term

Wanted	Equation for \(r\)	Working
Term in \(x^3\)
Term independent of \(x\)
The 5th term