Binomial Theorem summary

Key learning points

• Use the product form of coefficients for non-integer $n$.
• Truncate after the power of $x$ required.
• Factor out constants to reach a $(1+\cdots)^n$ shape.
• Always state $|x|<\ldots$ when using an infinite series.

Worked example 1 — Leading terms

First three terms of $(1-3x)^{-1}$ for $|3x|<1$.

Working

Here $n=-1$; coefficients are powers of 3.

Worked example 2 — Use the product form of coefficients for non-integer

Use this method from the Binomial Theorem sheet: Use the product form of coefficients for non-integer $n$.

Working

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

Worked example 3 — Truncate after the power of required

Use this method from the Binomial Theorem sheet: Truncate after the power of $x$ required.

Working

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

Worked example 4 — Factor out constants to reach a shape

Use this method from the Binomial Theorem sheet: Factor out constants to reach a $(1+\cdots)^n$ shape.

Working

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

Worked example 5 — Always state when using an infinite series

Use this method from the Binomial Theorem sheet: Always state $|x|<\ldots$ when using an infinite series.

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.