Factor and Remainder Theorems summary

Key learning points

• Remainder on division by $(x-a)$ is $f(a)$.
• $(x-a)$ is a factor iff $f(a)=0$.
• Find a root by inspection, then factorise the quadratic quotient.
• Include constant content factors in the final factorisation.

Worked example 1 — Factor theorem

Show $(x-2)$ is a factor of $f(x)=x^3-3x^2+4$.

Working

Zero remainder means a linear factor.

Worked example 2 — Remainder on division by is

Use this method from the Factor and Remainder Theorems sheet: Remainder on division by $(x-a)$ is $f(a)$.

Working

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

Worked example 3 — is a factor iff

Use this method from the Factor and Remainder Theorems sheet: $(x-a)$ is a factor iff $f(a)=0$.

Working

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

Worked example 4 — Find a root by inspection, then factorise the quadratic

Use this method from the Factor and Remainder Theorems sheet: Find a root by inspection, then factorise the quadratic quotient.

Working

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

Worked example 5 — Include constant content factors in the final factorisa

Use this method from the Factor and Remainder Theorems sheet: Include constant content factors in the final factorisation.

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.