The Chemistry Maths Book, Second Edition

2.3 Factorization and simplification of expressions 35

The inverse operation of factorization is usually called expansionor multiplying

out.

EXAMPLE 2.6Factorize:

(i) 2xy

2

1 − 14 x

2

y 1 + 16 xy

The expression 2 xyis a common factor. Therefore

2 xy

2

1 − 14 x

2

y 1 + 16 xy 1 = 1 (2xy) 1 × 1 (y) 1 − 1 (2xy) 1 × 1 (2x) 1 + 1 (2xy) 1 × 1 (3)= 12 xy(y 1 − 12 x 1 + 1 3)

(ii)x

2

1 − 15 x 1 − 16

The aim is to express the quadratic function as the product of two linear

functions; that is, to find numbers aand bsuch that

x

2

1 − 15 x 1 − 161 = 1 (x 1 + 1 a)(x 1 + 1 b)

Expansion of the product gives

(x 1 + 1 a)(x 1 + 1 b) 1 = 1 x(x 1 + 1 b) 1 + 1 a(x 1 + 1 b) 1 = 1 x

2

1 + 1 bx 1 + 1 ax 1 + 1 ab

and, therefore,

x

2

1 − 15 x 1 − 161 = 1 x

2

1 + 1 (a 1 + 1 b)x 1 + 1 ab

For this equation to be true for all values of xit is necessary that the coefficient

of each power of xbe the same on both sides of the equal sign:a 1 + 1 b 1 = 1 − 5 and

ab 1 = 1 −6.The two numbers whose sum is − 5 and whose product is 6 area 1 = 1 − 6

andb 1 = 11. Therefore

x

2

1 − 15 x 1 − 161 = 1 (x 1 − 1 6)(x 1 + 1 1)

(iii)x

2

1 − 19

Letx

2

1 − 191 = 1 (x 1 + 1 a)(x 1 + 1 b) 1 = 1 x

2

1 + 1 (a 1 + 1 b)x 1 + 1 ab. In this case there is no term linear

in x:a 1 + 1 b 1 = 10 , so thatb 1 = 1 −aandab 1 = 1 −a

2

1 = 1 − 9. Therefore and the

factorization is

x

2

1 − 191 = 1 (x 1 + 1 3)(x 1 − 1 3)

This is an example of the general formx

2

1 − 1 a

2

1 = 1 (x 1 + 1 a)(x 1 − 1 a).

(iv)x

4

1 − 15 x

2

1 + 14

The quartic in xis a quadratic in disguise. Replacement of x

2

by y, followed by

factorization gives

y

2

1 − 15 y 1 + 141 = 1 (y 1 − 1 1)(y 1 − 1 4)

a==± 93