Cambridge International Mathematics

Algebra (Expansion and factorisation) (Chapter 1) 53

The following procedure is recommended for factorising ax^2 +bx+c by splitting the middle term:

Step 1: Findac.

Step 2: Find the factors ofacwhich add tob.

Step 3: If these factors arepandq, replacebxby px+qx.

Step 4: Complete the factorisation.

Example 35 Self Tutor

Factorise 3 x^2 +17x+10:

For 3 x^2 +17x+10, 3 £10 = 30

We need to find two factors of 30 which have a sum of 17.

These are 2 and 15.

) 3 x^2 +17x+10=3x^2 +2x+15x+10

=x(3x+ 2) + 5(3x+2)

=(3x+ 2)(x+5)

Example 36 Self Tutor

Factorise 6 x^2 ¡ 11 x¡ 10 :

For 6 x^2 ¡ 11 x¡ 10 , 6 £¡10 =¡ 60

We need to find two factors of¡ 60 which have a

sum of¡ 11.

These are¡ 15 and 4.

) 6 x^2 ¡ 11 x¡10 = 6x^2 ¡ 15 x+4x¡ 10

=3x(2x¡5) + 2(2x¡5)

=(2x¡5)(3x+2)

EXERCISE 1L

1 Fully factorise:

a 2 x^2 +5x+3 b 2 x^2 +7x+5 c 7 x^2 +9x+2

d 3 x^2 +7x+4 e 3 x^2 +13x+4 f 3 x^2 +8x+4

g 8 x^2 +14x+3 h 21 x^2 +17x+2 i 6 x^2 +5x+1

j 6 x^2 +19x+3 k 10 x^2 +17x+3 l 14 x^2 +37x+5

2 Fully factorise:

a 2 x^2 ¡ 9 x¡ 5 b 3 x^2 +5x¡ 2 c 3 x^2 ¡ 5 x¡ 2

d 2 x^2 +3x¡ 2 e 2 x^2 +3x¡ 5 f 5 x^2 ¡ 14 x¡ 3

g 5 x^2 ¡ 8 x+3 h 11 x^2 ¡ 9 x¡ 2 i 3 x^2 ¡ 7 x¡ 6

As the product is negative, the

two numbers must be opposite

in sign. Since the sum is

negative, the larger number

must be negative.

IGCSE01

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Y:\HAESE\IGCSE01\IG01_01\053IGCSE01_01.CDR Wednesday, 10 September 2008 2:08:38 PM PETER