TMUA Binomial Expansion

The general term

$(a+b)^n=\sum_{k=0}^{n}\binom nk a^{n-k}b^k$, where $\binom nk=\dfrac{n!}{k!(n-k)!}$. The term with $b^k$ is $\binom nk a^{n-k}b^k$.

Finding one term

Write the power of $x$ in the general term, set it equal to the one you want, and solve for $k$. Remember that $a$ and $b$ include their own numbers and signs.

Products of brackets

For an expression like $(1+x)(1+2x)^4$, expand the power to the order you need and collect the matching terms.

Checks and shortcuts

A quick check is to substitute an easy value such as $x=1$ into the original expression and into your expansion. For $(2+x)^5$ the coefficients must add up to $3^5=243$.

Pascal's triangle gives $\binom nk$ for small $n$ without factorials: row 5 reads 1, 5, 10, 10, 5, 1. Use the symmetry $\binom nk=\binom n{n-k}$ to save effort, and be ready to simplify factorial expressions such as $\dfrac{7!}{5!}=42$.

Because there is no calculator, the numbers stay small. If a coefficient looks messy, recheck which term you have picked.

Worked examples

Common mistakes

• Forgetting to raise the numerical part of $b$ to the power $k$, for example $(2x)^3=8x^3$, not $2x^3$.
• Missing a sign when $b$ is negative.
• Taking only one route to the required power in a product of brackets.
• Using $\binom nk$ with the wrong $k$ for the power of $x$ asked for.

Test yourself on this topic with our free practice questions.