In (a+b)n, where n is a positive integer: T_r+1= nr an-rbr r=0,1,,n Collect the powers of x, then choose an integer r from 0 to n to give the required power. The term number is r+1. If no such integer exists, the requested power is absent. Example: (x+ 2x2)9, x 0 T_r+1= 9rx9-r(2x2)r= 9r2r,x9-3r **Wanted** | **Equation for r** | **Working** Term in x3 | 9-3r=3 | r=2 T_3= 9222x3=144x3 Term independent of x | 9-3r=0 | r=3 T_4= 9323=672 The 5th term | r+1=5 | r=4 T_5= 9424x-3=2016x-3 For subtraction, use (-b)r=(-1)rbr. The coefficient excludes the power of x.

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 an integer \(r\) from \(0\) to \(n\) to give the required power. The term number is \(r+1\). If no such integer exists, the requested power is absent.

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