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9.5 Factorising

9.5 Factorising

To expand a term with brackets, you multiply each term inside the brackets
by the term outside the brackets. 4(x + 3) = 4x + 12

When you factorise an expression you do the opposite.
You take the highest common factor and put it outside the brackets. 4x + 12 = 4(x + 3)

Worked example 9.5
Factorise these expressions. a 2x + 10 b 8 − 12y c 4a + 8ab d x − 5x
2
a 2x + 10 = 2(x + 5) The highest common factor of 2x and 10 is 2, so put the 2 outside the
brackets. Divide both terms by 2 and put the result inside the brackets.
Check the answer by expanding: 2 × x = 2x and 2 × 5 = 10.
b 8 − 12y = 4(2 − 3y) The highest common factor of 8 and 12y is 4, so put the 4 outside the
brackets. Divide both terms by 4 and put the result inside the brackets.
Check the answer by expanding: 4 × 2 = 8 and 4 × −3y = −12y.
c 4a + 8ab = 4a(1 + 2b) The highest common factor of 4a and 8ab is 4a, so put the 4a outside the
brackets. Divide both terms by 4a and put the result inside the brackets.
Check the answer: 4a × 1 = 4a and 4a × 2b = 8ab.
2
2
d x − 5x = x(x − 5) The highest common factor of x and 5x is x, so put the x outside the
brackets. Divide both terms by x and put the result inside the brackets.
2
Check the answer: x × x = x and x × −5 = −5x.

) Exercise 9.5

1 Copy and complete these factorisations.
a 3x + 6 = 3(x + ) b 10y − 15 = 5(2y − ) c 6xy + 12y = 6y(x + )
d 4x + x = x(4x + ) e 9 − 12y = 3( − ) f 2y − 7y = y( − )
2
2
2 Factorise each of these expressions.
a 2x + 4 b 3y − 18 c 10z + 5 d 8a − 4 e 4b + 6 f 16n − 20
g 10 − 5x h 14 + 21x i 8 − 10y j 18 + 24z k 9 + 15m l 30 − 20k

3 Factorise each of these expressions.
a 3x + x b 6y − 12y c z + 4z d 4a − 2a e 3b + 9b f 12n − 15n 2
2
2
2
2
2
g 18y − 9x h 12y + 9x i 8xy − 4y j 15z + 10yz k 14m + 6mn l 26k − 13kp
4 Copy and complete these factorisations.
a 2x + 6y + 8 = 2(x + 3y + ) b 4y − 8 + 4x = 4(y − + x) c 9xy + 12y − 15 = 3(3xy + − 5)
2
2
d 5x + 2x + xy = x(5x + + ) e 9y − y − xy = y( − − ) f 3y − 9y + 6xy = 3y( − + )
2
5 Read what Tanesha says.
Show that she is right. When I expand 5(2x + 6) + 2(3x − 5), then collect like terms
and fi nally factorise the result, I get the expression 4(4x + 5).

6 Read what Shen says. When I expand 6(3y + 2) − 4(y − 2), then collect like terms
Show that he is wrong. and fi nally factorise the result, I get the expression 2(7y + 2).
Explain the mistake he
has made.
9 Expressions and formulae 91