Cambridge International Mathematics

Algebra (Expansion and factorisation) (Chapter 1) 39

In this section we expand more complicated expressions by repeated use of the expansion laws.

Consider the expansion of (a+b)(c+d+e):

Now (a+b)(c+d+e)

=(a+b)c+(a+b)d+(a+b)e

=ac+bc+ad+bd+ae+be

Compare: ¤(c+d+e)

=¤c+¤d+¤e

Notice that there are 6 terms in this expansion and that each term within the first bracket is multiplied by

each term in the second.

2 terms in the first bracket £ 3 terms in the second bracket 6 terms in the expansion.

Example 12 Self Tutor

Expand and simplify: (x+ 3)(x^2 +2x+4)

(x+ 3)(x^2 +2x+4)

=x(x^2 +2x+4)+3(x^2 +2x+4)

=x^3 +2x^2 +4x fall terms in the 2 nd bracket£xg

+3x^2 +6x+12 fall terms in the 2 nd bracket£ 3 g

=x^3 +5x^2 +10x+12 fcollecting like termsg

Example 13 Self Tutor

Expand and simplify: (x+2)^3

(x+2)^3

=(x+2)£(x+2)^2

=(x+ 2)(x^2 +4x+4)

=x^3 +4x^2 +4x fall terms in the 2 nd bracket£xg

+2x^2 +8x+8 fall terms in the 2 nd bracket£ 2 g

=x^3 +6x^2 +12x+8 fcollecting like termsg

Example 14 Self Tutor

Expand and simplify:

a x(x+ 1)(x+3) b (x+ 1)(x¡3)(x+2)

a x(x+ 1)(x+3)

=(x^2 +x)(x+3) fall terms in the first bracket£xg

=x^3 +3x^2 +x^2 +3x fexpanding the remaining factorsg

=x^3 +4x^2 +3x fcollecting like termsg

E FURTHER EXPANSION [2.7]

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Y:\HAESE\IGCSE01\IG01_01\039IGCSE01_01.CDR Thursday, 11 September 2008 9:05:08 AM PETER