Home > Subjects > Mathematics > Level 2 > 2.6 Algebra > Subject content > Factorising algebraic expressions

- Subject: Mathematics
- AS: 91261
- Level: 2
- Credits: 4
- External

## Mathematics 2.6 Apply algebraic methods in solving problems

### Factorising algebraic expressions

Factorise is the reverse of expand i.e. put in brackets.

To factorise, rewrite the expression as factors multiplied together.

#### Two terms

Look for the largest number that will divide into all terms and any common letters.

e.g. 12x + 20xy = 4x(3 + 5y) This is because 4x is the largest factor of both 12x and 20xy.

Check by expanding your answer.

See online examples of common factors.

#### Two terms: Difference of two squares

Look for two square terms that have been subtracted.

e.g. 4x^{2}– 9 = (2x)^{2}– 3^{2}= (2x + 3)(2x – 3)

See online examples of factoring the difference of squares.

#### Three terms

Look for a common factor first and then factorise further.

e.g. 3x^{2}– 6x – 15 = 3(x^{2}– 2x – 5)

If no common factor, look for two numbers that multiply to give the number by itself (constant term) and add to give the number in front of x.

e.g. x^{2}– 2x - 15 = (x + 3)(x – 5) This is because –5 times 3 = –15 and –5 plus 3 = –2

Check

Hence 3x^{2}– 6x – 45 = 3(x + 3)(x – 5)

If there is no number written in front of x remember it is really a 1.

e.g. x^{2}+ x – 6 = (x + 3)(x – 2) This is because 3 times -2 = –6 and 3 plus –2 = 1

See notes on:

#### Harder factorising

If there is a number in front of x^{2} that is not a common factor use guess and check or the following method.

Two numbers that multiply to give –6 and add to give +5 are +6 and –1. Write 5x as 6x - x.

e.g. 2x^{2}+ 5x – 3= 2x^{2}+ 6x – x – 3

When there are four terms, factorise in pairs first.

e.g. 2x^{2}+ 6x – x – 3 = 2x(x + 3) – 1(x + 3)

Take out the common bracket (x + 3).

e.g. 2x(x + 3) – 1(x + 3) = (x + 3)(2x – 1)

See notes on:

- Factoring quadratics: The hard case
- Algebasics – Section 8: Factoring (Macromedia Flash® and sound required)