The Binomial Theorem summary

Key learning points

• $(a+b)^n=\sum \binom{n}{k}a^{n-k}b^k$ for $n\in\mathbb{N}$.
• For $|x|<1$, $(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\cdots$.
• Rewrite expressions as $A(1+\cdots)^n$ before expanding.
• State the validity range for infinite expansions.

Worked example 1 — Expand

Expand $(1+2x)^3$.

Working

Coefficients 1,3,3,1 with powers of $2x$.

Worked example 2 — for

Use this method from the The Binomial Theorem sheet: $(a+b)^n=\sum \binom{n}{k}a^{n-k}b^k$ for $n\in\mathbb{N}$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 3 — For ,

Use this method from the The Binomial Theorem sheet: For $|x|<1$, $(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\cdots$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 4 — Rewrite expressions as before expanding

Use this method from the The Binomial Theorem sheet: Rewrite expressions as $A(1+\cdots)^n$ before expanding.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 5 — State the validity range for infinite expansions

Use this method from the The Binomial Theorem sheet: State the validity range for infinite expansions.

Working

Each key learning point matches a question type on the printable worksheet.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.