Cambridge International Mathematics

32 Algebra (Expansion and factorisation) (Chapter 1)

Consider the expression 2(x+3). We say that 2 is thecoefficientof the expression in the brackets. We

canexpandthe brackets using thedistributive law:

a(b+c)=ab+ac

The distributive law says that we must multiply the coefficient by each term within the brackets, and add

the results.

Geometric Demonstration:

The overall area is a(b+c).

However, this could also be found by adding the areas of

the two small rectangles: ab+ac.

So, a(b+c)=ab+ac: fequating areasg

Example 1 Self Tutor

Expand the following:

a 3(4x+1) b 2 x(5¡ 2 x) c ¡ 2 x(x¡3)

a 3(4x+1)

= 3 £ 4 x+ 3 £ 1

=12x+3

b 2 x(5¡ 2 x)

=2x(5 +¡ 2 x)

= 2 x£5+ 2 x£¡ 2 x

=10x¡ 4 x^2

c ¡ 2 x(x¡3)

=¡ 2 x(x+¡3)

=¡ 2 x£x+¡ 2 x£¡ 3

=¡ 2 x^2 +6x

With practice, we do not need to write all of these steps.

Example 2 Self Tutor

Expand and simplify:

a 2(3x¡1) + 3(5¡x) b x(2x¡1)¡ 2 x(5¡x)

a 2(3x¡1) + 3(5¡x)

=6x¡2+15¡ 3 x

=3x+13

b x(2x¡1)¡ 2 x(5¡x)

=2x^2 ¡x¡ 10 x+2x^2

=4x^2 ¡ 11 x

EXERCISE 1A

1 Expand and simplify:

a 3(x+1) b 2(5¡x) c ¡(x+2) d ¡(3¡x)

e 4(a+2b) f 3(2x+y) g 5(x¡y) h 6(¡x^2 +y^2 )

A THE DISTRIBUTIVE LAW [2.7]

Notice in that the minus

sign in front of affects

terms inside the

following bracket.

2 x

both

b

b c

a

bc+

IGCSE01

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Y:\HAESE\IGCSE01\IG01_01\032IGCSE01_01.CDR Wednesday, 10 September 2008 2:03:42 PM PETER