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Factorising expressions

To factorise means to write an expression as a product. We first take out the highest common factor: 6x + 15 = 3(2x + 5). Sometimes the difference of two squares fits: x² − 25 = (x + 5)(x − 5). We check by multiplying out.

Year 9 and upper secondary · Updated

Watch Fox work it out on the board

Fox works one step at a time. Press play.

Take out a common factor

We want to find what is in every term and put it in front of a bracket.

  • Take the largest number that divides all the coefficients.
  • Take the lowest power of x that is in every term.
  • Divide every term by the factor. What is left goes inside the bracket.

8 and 12 are both divisible by 4, and both terms contain x. The factor is 4x.

8x² + 12x = 4x(2x + 3)

The squaring rules in reverse

If the first and last terms are squares, we check the middle term. Here 25 = 5² and 2 · x · 5 = 10x.

x² + 10x + 25 = (x + 5)²

With a minus in the middle, it is the rule for squaring a difference.

x² − 6x + 9 = (x − 3)²

The difference of two squares

A square minus a square becomes two brackets with different signs.

x² − 49 = (x + 7)(x − 7)

Example

Example 1: 10x² − 15x

We want to take out the largest factor that is in both terms.

  1. 10 and 15 are divisible by 5. Both terms contain x. The factor is 5x.

    5x

  2. We divide every term by 5x: 10x² ÷ (5x) = 2x and −15x ÷ (5x) = −3.

    5x(2x − 3)

  3. 2x and 3 have no common factor left, so we are done.

    10x² − 15x = 5x(2x − 3)

We expand the bracket: 5x · 2x = 10x² and 5x · (−3) = −15x. That is correct.

Example 2: 3x² − 75

We want to factorise as far as possible. First a common factor, then a pattern.

  1. We take out 3: 3x² ÷ 3 = x² and 75 ÷ 3 = 25.

    3(x² − 25)

  2. x² − 25 is a square minus a square, because 25 = 5². That is a difference of two squares.

    3(x + 5)(x − 5)

  3. No factor can be split any further.

    3x² − 75 = 3(x + 5)(x − 5)

With x = 1, the start gives 3 − 75 = −72 and the answer gives 3 · 6 · (−4) = −72. That is correct.

Pitfalls

  • Not taking out the largest factor, so that 12x + 18 becomes 2(6x + 9).

    6x and 9 still share the factor 3. The highest common factor is 6: 6(2x + 3).

  • Using the difference of two squares on a sum, so that x² + 16 becomes (x + 4)(x − 4).

    (x + 4)(x − 4) = x² − 16. The rule only works for a square minus a square.

Practise on your own

0 of 4 correct

  1. Factorise 7x + 21.

  2. Factorise 6x² − 9x fully.

  3. Factorise x² + 12x + 36.

  4. Factorise 2x² − 32 fully.

Common questions

Why do we factorise?

One reason is to solve equations. x² − 5x = 0 can be written x(x − 5) = 0. A product is zero only if one of its factors is zero, so x = 0 or x = 5.

Can every expression be factorised?

No. x² + 16 cannot be written as two brackets using real numbers. Then we leave the expression as it is.

How do I check a factorisation?

Multiply out again. If we get back the expression we started with, it is correct.

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