How Do You Expand (a + b)^n Using the Binomial Theorem?

Use the binomial theorem:

$$(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r,$$

where $n$ is a non-negative integer. The formula produces $n+1$ terms whose coefficients can be found using Pascal's triangle or $\binom{n}{r}$.

How the expansion works

The general term is

$$T_{r+1}=\binom{n}{r}a^{n-r}b^r, \qquad \binom{n}{r}=\frac{n!}{r!(n-r)!}.$$

Begin with $r=0$, then increase $r$ by one until $r=n$. As this happens, the exponent of $a$ decreases from $n$ to , while the exponent of increases from to . In every term, the two exponents add to .

For example, expand $(2x-3)^4$. Substitute $a=2x$, $b=-3$, and :

$$(2x-3)^4=\sum_{r=0}^{4}\binom{4}{r}(2x)^{4-r}(-3)^r.$$

The fourth row of Pascal's triangle gives the coefficients $1,4,6,4,1$. Therefore,

$$\begin{aligned} (2x-3)^4 &=(2x)^4+4(2x)^3(-3)+6(2x)^2(-3)^2\\ &\quad+4(2x)(-3)^3+(-3)^4\\ &=16x^4-96x^3+216x^2-216x+81. \end{aligned}$$

You can also use the general term to select one coefficient. In the expansion above, the term in $x^2$ occurs when $4-r=2$, so $r=2$. Its coefficient is therefore $$. This avoids writing all five terms and is usually the fastest method when a question asks you to find a coefficient rather than expand the whole expression.

Check the result systematically: there should be five terms before like terms are combined, the powers of $x$ should descend from $4$ to $0$, and each unsimplified term should have total degree $4$. Keeping $(-3)^r$ in brackets prevents sign errors.

A common misconception is that binomial signs always alternate. They alternate only when one binomial term is negative, as in $(a-b)^n$, because odd powers of the negative term are negative and even powers are positive. For $(a+b)^n$, both terms are positive, so every term is positive.

IB exam technique

This is IB Mathematics: Analysis and Approaches, SL 1.9, studied by both SL and HL students. It can be tested on Paper 1 or Paper 2, although exact expansions are especially suitable for Paper 1. Show substitution into the general term before simplifying. If asked for one coefficient, equate the required power with the relevant exponent and find $r$ without expanding the entire binomial. Remember that $T_{r+1}$ is the $(r+1)$th term, so the term corresponding to is the fourth term.