Unit 11 Section 3 : Factorising

In this section we consider examples of the process of factorising, whereby the process of removing brackets is reversed and brackets are introduced into expressions.

Example 1

Factorise:

(a)

8x + 12

Note that both terms are multiples of 4, so we can write,

8x + 12 = 4(2x + 3)

(b)

35x + 28

Here both terms are multiples of 7, so

35x + 28 = 7(5x + 4)

Results like this can be checked by multiplying out the bracket to get back to the original expression.

Example 2

Factorise,

(a)

x^2 + 2x

Here, as both terms are multiples of x, we can write,

x^2 + 2x = x(x + 2)

(b)

3x^2 - 9x

In this case, both terms are multiples of x and 3, giving,

3x^2 - 9x = 3x(x - 3)

(c)

x^3 - x^2

In this example, both terms are multiples of x2 ,

x^3 - x^2 = x^2 (x - 1)

Sometimes it is possible to factorise in stages. For example, in part (b), you could have worked like this:
3x^2 - 9x = 3(x^2 - 3x)
= 3x(x - 3)

Example 3

Factorise:

(a)

x^2 + 9x + 18

This expression will need to be factorised into two brackets:

x^2 + 9x + 18 = (x       )(x       )

As the expression begins x2, both brackets must begin with x. The two numbers to go in the brackets must multiply together to give 18 and add to give 9. So they must be 3 and 6, giving,

x^2 + 9x + 18 = (x + 3)(x + 6)

You can check this result by multiplying out the brackets.

(b)

x^2 + 2x - 15

We note first that two brackets are needed and that both must contain an x, as shown:

x^2 + 2x - 15 = (x       )(x       )

Two other numbers are needed which, when multiplied give –15 and when added give 2. In this case, these are –3 and 5. So the factorisation is,

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

Check this result by multiplying out the brackets.

(c)

x^2 - 7x + 12

Again, we begin by noting that,

x^2 - 7x + 12 = (x       )(x       )

We require two numbers which, when multiplied give 12 and when added give –7. In this case, these numbers are –3 and –4.

x^2 - 7x + 12 = (x - 3)(x - 4)

Exercises

Note: To type indeces on this page use ^ sign. e.g.  n2
Question 1

Factorise:

(a)4x - 2
(b)6x - 12
(c)5x - 20
(d)4x + 32
(e)6x - 8
(f)8 - 12x
(g)21x - 14
(h)15x + 20
(i)30 - 10x
Question 2

Factorise:

(a)x^2 + 4x
(b)x^2 - 3x
(c)4x - x^2
(d)6x^2 + 8x
(e)9x^2 + 15x
(f)7x^2 - 21x
(g)28x - 35x^2
(h)6x^2 - 14x
(i)5x^2 - 3x
Question 3

Factorise:

(a)x^3 + x^2
(b)2x^2 - x^3
(c)4x^3 - 2x^2
(d)8x^3 + 4x^2
(e)16x^2 - 36x^3
(f)4x^3 + 22x^2
(g)16x^2 - 6x^3
(h)14x^3 + 21x^2
(i)28x^3 - 49x^2
Question 4

(a)

Expand (x + 5)(x - 5).

(b)

Factorise x^2 - 25.

(c)

Factorise each of the following:

(i) x^2 - 49
(ii) x^2 - 64
(iii) x^2 - 100
(iv) x^2 - a^2
(v) x^2 - 4b^2
Question 5

Factorise:

(a)x^2 + 7x + 12
(b)x^2 + 8x + 7
(c)x^2 + 11x + 18
(d)x^2 + 12x + 27
(e)x^2 + 17x + 70
(f)x^2 + 6x + 8
(g)x^2 + 16x + 28
(h)x^2 + 18x + 77
(i)x^2 + 16x + 63
Question 6

Factorise:

(a)x^2 + x - 2
(b)x^2 + x - 20
(c)x^2 - x - 12
(d)x^2 - 13x + 36
(e)x^2 - 10x + 16
(f)x^2 + x - 42
(g)x^2 + 13x - 30
(h)x^2 - 17x + 72
(i)x^2 - 2x - 99
Question 7

The area of the rectangle shown is

x^2 - 5x.

Express a in terms of x.

a =
Question 8

The area of the rectangle shown is

x^2 + 11x + 30.

Express a in terms of x.

a =
Question 9

The area of the triangle shown is

x^2 + x - 5.

Express h in terms of x.

h =
Question 10

The area of the trapezium shown is

x^2 + 10x + 18.

Determine a.

a =