Binomial Theorem (P4) Topic Review summary

Key learning points

• Positive integer $n$: full expansion with binomial coefficients.
• General binomial series for rational $n$ with $|x|<1$ when taught.
• Rewrite as $A(1+\\cdots)^n$ before expanding.
• Find particular coefficients without full expansion.
• State validity for infinite expansions.

Worked example 1 — Integer

Coefficient of $x^2$ in $(1+x)^5$.

Working

$\\binom{5}{2}$.

Worked example 2 — Form

Prepare $\\frac{{1}}{{\\sqrt{{4+x}}}}$ for expansion.

Working

Factor out 4.

Worked example 3 — Positive integer n: full expansion with bi

Positive integer $n$: full expansion with binomial coefficients.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — General binomial series for rational n wit

General binomial series for rational $n$ with $|x|<1$ when taught.

Working

Attempt without notes, then check the answers PDF.

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.