7.3PureStretch

The Factor Theorem

The factor theorem is a shortcut for finding factors of polynomials without long division. If substituting a value makes the polynomial zero, you have found a factor — and that is the key to factorising cubics.

30 min Video by Zeeshan Zamurred Algebraic Methods

Edexcel AS Level Maths: 7.3 The Factor TheoremWatch the full walkthrough before the notes below.

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What you'll be able to do

State and use the factor theorem
Test whether a linear expression is a factor
Find a first factor by trying small values
Fully factorise a cubic polynomial

The theorem

The factor theorem says: if $f (p) = 0$ , then $(x - p)$ is a factor of $f (x)$ — and the reverse is also true. So testing a value tells you instantly whether a bracket divides the polynomial.

f (p) = 0 ⟺ (x - p) is a factor of f (x)

Substitute

x = p

; a zero output means

(x - p)

is a factor.

1Substitute

x = 1

1 - 2 - 1 + 2 = 0

2Since

f (1) = 0

(x - 1)

is a factor.

Answer

(x - 1)

is a factor

Tip — For (x − p), substitute x = p. For a factor like (2x − 1), test x = ½.

Finding the first factor

To start factorising a cubic, try small values such as $\pm 1, \pm 2$ (and factors of the constant term) until one gives zero. That gives your first bracket.

Fully factorising a cubic

Once you have one factor, divide it out (long division) to get a quadratic, then factorise that quadratic normally. The result is the cubic as a product of three linear factors.

f (1) = 0

, so

(x - 1)

is a factor.

2Divide:

x^{3} - 2 x^{2} - x + 2 = (x - 1) (x^{2} - x - 2)

3Factorise the quadratic:

x^{2} - x - 2 = (x - 2) (x + 1)

Answer

(x - 1) (x - 2) (x + 1)

Tip — Constant term is a goldmine: the roots are usually among its factors (here ±1, ±2).

Formula recap

f (p) = 0 \Rightarrow (x - p) is a factor

The factor theorem.

f (\frac{b}{a}) = 0 \Rightarrow (a x - b) is a factor

For non-unit factors.

Common mistakes to avoid

Testing x = −p for the factor (x − p).

For (x − p) substitute x = +p. The sign flips.

Stopping after finding one factor of a cubic.

Divide it out and factorise the resulting quadratic for the full factorisation.

Key takeaways

f(p) = 0 ⇔ (x − p) is a factor.
Test small values (factors of the constant term) to find the first factor.
Divide out that factor, then factorise the remaining quadratic.

Test yourself

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