Factorising

Video lesson6 worked examples

Theory

Factorising is the inverse process to expanding brackets.

You must be able to identify and extract a common factor, factorise the difference of two squares (e.g., $p^2x^2 - a^2$ ), and factorise trinomials with both unitary and non-unitary $x^2$ coefficients.

Some expressions will require a combination of these methods (e.g., extracting a common factor before factorising a difference of squares).

The Golden Rule: always check for a common factor first — take it outside a bracket before trying anything else. Then look at what is left: two terms is likely a difference of two squares; three terms is a trinomial.

⚠️ Common Examiner Traps

Missing the common factor: jumping straight to a trinomial and forgetting to take out a factor first leaves the answer only partly factorised — and loses marks.
Sum of two squares: $x^2 + 9$ does not factorise. The difference of two squares needs a minus sign.
Signs in a trinomial: the two numbers must multiply to the last term and add to the middle one — check both, and mind the signs.
Not factorising fully: after the first step, always ask whether what remains can be factorised again.

Worked examples

Example 1

Common Factor

Factorise $6x^2 + 9x$ .

Step 1: Find the highest common factor of both terms. $6x^2$ and $9x$ share $3x$ .

Step 2: Take $3x$ outside the bracket and write what is left inside:

Answer: $3x(2x + 3)$ .

Example 2

Difference of Two Squares

Factorise $9m^2 - 25$ .

Step 1: Both terms are perfect squares with a minus between them. Identify the square roots: $\sqrt{9m^2} = 3m$ and $\sqrt{25} = 5$ .

Step 2: Write as one bracket plus, one bracket minus:

Answer: $(3m - 5)(3m + 5)$ .

Example 3

Trinomial (unitary)

Factorise $x^2 - x - 12$ .

Step 1: Find two numbers that multiply to $-12$ and add to $-1$ . The pair $-4$ and $+3$ works: $-4 \times 3 = -12$ and $-4 + 3 = -1$ .

Answer: $(x - 4)(x + 3)$ .

Example 4

Trinomial (non-unitary)

Factorise $5x^2 - 13x + 6$ .

Step 1: With a coefficient on $x^2$ , look for brackets whose first terms multiply to $5x^2$ and whose last terms multiply to $+6$ , chosen so the cross-terms add to $-13x$ .

Step 2: Testing $(5x - 3)(x - 2)$ : the cross-terms are $-10x - 3x = -13x$ ✓.

Answer: $(5x - 3)(x - 2)$ .

Example 5

Rearranging first (negative $x^2$ term)

Factorise $15 + 2x - x^2$ .

Step 1: A trinomial is easiest to factorise when the $x^2$ term is positive and written first. Reorder, then take out a factor of $-1$ :

15 + 2x - x^2 = -x^2 + 2x + 15 = -(x^2 - 2x - 15)

Step 2: Factorise the bracket as usual — two numbers multiplying to $-15$ and adding to $-2$ are $-5$ and $+3$ :

-(x^2 - 2x - 15) = -(x - 5)(x + 3)

Answer: $-(x - 5)(x + 3)$ .

Example 6

🔗 Bringing it together

Factorise fully $3y^2 - 27$ .

Step 1: Following the Golden Rule, take out the common factor of 3 first: $3(y^2 - 9)$ .

Step 2: The bracket is now a difference of two squares, so factorise it again: $y^2 - 9 = (y - 3)(y + 3)$ .

Step 3: Combine — the word “fully” is the signal that a second step was needed:

Answer: $3(y - 3)(y + 3)$ .

Ready to practise?

14 Factorising questions with answers and video solutions