Additional Maths Notes

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make notes on the factor theorem

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The Factor Theorem

The Factor Theorem says: if f(a) = 0 for a polynomial f(x), then (x − a) is a factor of f(x).

Why it's useful

Lets you find factors of a cubic (or higher) polynomial without guessing and dividing repeatedly.
Combined with the Remainder Theorem (f(a) gives the remainder when f(x) is divided by (x − a)) to fully factorise a polynomial.
Try values of a that divide the constant term first — these are the most likely roots.

Worked example

Question: Show that (x − 2) is a factor of f(x) = x³ − 3x² − 4x + 12, then fully factorise f(x).

Answer: f(2) = 8 − 12 − 8 + 12 = 0, so by the Factor Theorem (x − 2) is a factor. Dividing f(x) by (x − 2) gives x² − x − 6, which factorises to (x − 3)(x + 2). So f(x) = (x − 2)(x − 3)(x + 2).

Common mistake: stopping after finding one factor and one quadratic, without factorising the quadratic itself the rest of the way — a fully factorise question expects linear factors only.

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