Factor and Remainder Theorems Topic Review summary

Key learning points

• Factor theorem: $(x-a)$ is a factor of $f(x)$ iff $f(a)=0$; for $(bx-a)$ test $f\!\left(\frac{a}{b}\right)=0$.
• Remainder theorem: remainder on division by $(x-a)$ is $f(a)$.
• Algebraic division gives the quotient and remainder; check by expanding.
• Given one root, factor out the linear factor and solve the quadratic quotient.
• Unknown coefficients: equations from $f(a)=0$ and/or a given remainder.

Worked example 1 — Factor check

Show $(x-1)$ factors $x^3-2x^2-11x+12$.

Working

Zero value ⇒ exact factor.

Worked example 2 — Unknown

$(x+2)$ factors $x^3-2x^2+ax+6$. Find $a$.

Working

$f(-2)=0$.

Worked example 3 — Practice point 1

Factor theorem: $(x-a)$ is a factor of $f(x)$ iff $f(a)=0$; for $(bx-a)$ test $f\\!\\left(\\frac{a}{b}\\right)=0$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Practice point 2

Remainder theorem: remainder on division by $(x-a)$ is $f(a)$.

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.