Cambridge International Mathematics

Algebra (Expansion and factorisation) (Chapter 1) 49

Aquadratic trinomialis an algebraic expression of the form ax^2 +bx+c wherexis a variable anda,

b,care constants, a 6 =0:

In this exercise we will look at quadratic trinomials for which a=1. They have the form x^2 +bx+c.

Consider the expansion of the product (x+ 2)(x+5):

(x+ 2)(x+5)=x^2 +5x+2x+2£ 5 fusing FOILg

=x^2 +[5+2]x+[2£5]

=x^2 +[sumof 2 and5]x+[productof 2 and5]

=x^2 +7x+10

In general, x^2 +(®+ ̄)x + ® ̄ =(x+®)(x+ ̄)

the coefficient of

xis thesumof

®and ̄

the constant term is

theproductof®

and ̄

So, to factorise x^2 +7x+10into (x+::::::)(x+::::::), we seek two numbers which add to 7 , and when

multiplied give 10.

These numbers are +2 and +5,sox^2 +7x+10=(x+ 2)(x+5)

Example 30 Self Tutor

Factorise: x^2 +11x+24

We need to find two numbers which have sum=11and product=24.

Pairs of factors of 24 :

Factor product 1 £ 24 2 £ 12 3 £ 8 4 £ 6

Factor sum 25 14 11 10

The numbers we want are 3 and 8.

) x^2 +11x+24=(x+ 3)(x+8)

With practice, you should be able to perform factorisations like this in your head.

Example 31 Self Tutor

Factorise: x^2 ¡ 7 x+12

sum=¡ 7 and product=12

) the numbers are¡ 3 and¡ 4

) x^2 ¡ 7 x+12=(x¡3)(x¡4)

FACTORISING x

2

K +bx+c [2.8]

Most of the time we

can find the two

numbers mentally.

The sum is negative but the

product is positive, so both

numbers must be negative.

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Y:\HAESE\IGCSE01\IG01_01\049IGCSE01_01.CDR Wednesday, 10 September 2008 2:08:17 PM PETER