Partial Fractions Topic Review summary

Key learning points

• Make proper by dividing if needed.
• Linear factors $\frac{A}{x-a}+\frac{B}{x-b}$; repeated factors need every power.
• Irreducible quadratics need a linear numerator $Dx+E$.
• Equate coefficients or substitute clever $x$ values.
• Integrate or expand each piece after decomposing.

Worked example 1 — Two linears

$\frac{5x+1}{(x-1)(x+2)}$.

Working

Cover-up / equate.

Worked example 2 — Repeated

Include $\frac{B}{(x-a)^2}$ for a repeated linear factor.

Working

Missing the squared term loses marks.

Worked example 3 — Make proper by dividing if needed

Apply this idea from the Topic Review: Make proper by dividing if needed.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Linear factors \A{x-a}+\B{x-b}; repeated f

Linear factors $\\frac{A}{x-a}+\\frac{B}{x-b}$; repeated factors need every power.

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.