4.5Factorisation

Factorisation is the reverse of expansion. Instead of distributing a factor into a bracket, we extract a common factor out of the terms.

Factorisation

\displaystyle ab + ac = a(b + c)

Numerical common factor

Factorise $\displaystyle 15y - 21$ completely.

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HCF of

15

and

21

3

15y - 21 = 3(5y - 7)

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Factorise $4a + 4b$ completely.

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HCF of

4

and

4

4

4a + 4b = 4(a + b)

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Factorise $\displaystyle 12mn^2 - 8m^2n$ completely.

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Common factors: HCF of

12

and

8

4

; both terms contain

m

and

n

HCF

= 4mn

12mn^2 - 8m^2n = 4mn(3n - 2m)

Factorise $\displaystyle -18bx + 12by$ completely.

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Write the positive term first:

-18bx + 12by = 12by - 18bx

HCF of

12

and

18

6

, and both terms contain

b

, so the common factor is

6b

12by - 18bx = 6b(2y - 3x)

Now you try

Factorise $-4x + 6y$ completely.

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-4x + 6y = 6y - 4x \quad \scriptsize\textit{rearrange order of terms}

HCF of

4

and

6

2

6y - 4x = 2(3y - 2x)

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(a) Factorise $\displaystyle 8a^2 + 12a$ completely. (b) Hence evaluate $\displaystyle 8a^2 + 12a$ when $\displaystyle a = 50$ .

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(a) HCF of

8

and

12

4

; both terms contain

a

, so the HCF is

4a

8a^2 + 12a = 4a(2a + 3)

(b) Substitute

a = 50

\displaystyle 4a(2a + 3)

\displaystyle = 200 \times 103

\displaystyle = 20\,600

The factorised form turns the sum into one easy product.