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CHAPTER 7
FACTORS
AIM: To use factorisation in simplifying expressions and equations.
7.1 FACTORISATION
It is a technique used to simplify mathematical expressions and equations.
EXAMPLE 7.1
The expression [2 × 52 - 10] is the same as, or equal to 40.
EXAMPLE 7.2
EXAMPLE 7.3
The general rule in simplification is to factorise an expression into the simplest factors possible.
To do this, we can identify specific types of expressions and follow a fixed procedure to factorise
each one of them.
7.2 THE COMMON FACTOR
EXAMPLE 7.4
ab - abc + ac
This expression consists out of three terms, ie. ab, abc and ac. There is a common factor a.
ab - abc + ac
= a(b - bc + c)
The expression with three terms has been factorised and consists out of only one term
a(b - bc + c).
EXERCISE 7.1
Factorise the following expressions:
7.1.1 ab + 2a 7.1.2 B r2h + 2Br 7.1.3 36x4 - 24x2
7.1.4 12a2 - 6ab + 9b2 7.1.5 x(x + y) - y(x + y) 7.1.6 (c + d) - a(d + c)
7.1.7 4t(q + r) - 6(r + q) 7.1.8 (a - 3)x2 - 2x(a - 3) - (a - 3) 7.1.9 (q2 - r) - b(r - q2)
7.1.10 4(x - y) - 6c(x - y)
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7.3 QUADRATIC TRINOMIALS
7.3.1 The product of (x + 2)(x + 3) = x2 + 5x + 6.
This was determined as follows:
First term: x × x = x2
Middle term: 2 × x + 3 × x = 5x
Third term: 2 × 3 = 6
The inverse process is called factorisation. The expression x2 + 5x + 6 factors can be determined
as follows:
(1) (x ) (x )
(2) (x + ) (x + )
(3) Consider the following possible factors of 6 to determine the sum of +5x for the
midterm and a product of +6 for the third term:
6=6×1 or 6=2x3
and 6x + 1x = 7x or 2x + 3x = 5x
We sum because the sign of the third term is +.
Therefore to resolve into factors x2 + 5x + 6 = (x + 2)(x + 3).
7.3.2 Similarly the product of (x + 2)(x - 4) = x2 - 2x - 8. To factorise x2 - 2x - 8 follow
the steps.
(1) (x )(x )
(2) (x + )(x - )
(3) Consider the following possible factors of 8 to determine the difference of -2x
for the midterm and a product of -8 for the third term:
8=1x8 or 8=2x4
and 1x - 8x = -7x or 2x - 4x = -2x
We subtract because the sign of the third term is -.
To resolve into factors x2 - 2x - 8 = (x + 2)(x - 4).
7.3.3 The square of a trinomial
A special trinomial expression is when both factors are the same.
The product of (x - 3)(x - 3) = x2 - 6x + 9. Look at the following properties:
(1) The coefficient of the first and third terms is a square.
(2) The coefficient of the midterm is two times the product of the square roots of the
first and third terms.
Consider: First term Third term Midterm
12 = 1 32 = 9
To resolve into factors x2 - 6x + 9 = (x - 3)2.
CHAPTER 7
FACTORS
AIM: To use factorisation in simplifying expressions and equations.
7.1 FACTORISATION
It is a technique used to simplify mathematical expressions and equations.
EXAMPLE 7.1
The expression [2 × 52 - 10] is the same as, or equal to 40.
EXAMPLE 7.2
EXAMPLE 7.3
The general rule in simplification is to factorise an expression into the simplest factors possible.
To do this, we can identify specific types of expressions and follow a fixed procedure to factorise
each one of them.
7.2 THE COMMON FACTOR
EXAMPLE 7.4
ab - abc + ac
This expression consists out of three terms, ie. ab, abc and ac. There is a common factor a.
ab - abc + ac
= a(b - bc + c)
The expression with three terms has been factorised and consists out of only one term
a(b - bc + c).
EXERCISE 7.1
Factorise the following expressions:
7.1.1 ab + 2a 7.1.2 B r2h + 2Br 7.1.3 36x4 - 24x2
7.1.4 12a2 - 6ab + 9b2 7.1.5 x(x + y) - y(x + y) 7.1.6 (c + d) - a(d + c)
7.1.7 4t(q + r) - 6(r + q) 7.1.8 (a - 3)x2 - 2x(a - 3) - (a - 3) 7.1.9 (q2 - r) - b(r - q2)
7.1.10 4(x - y) - 6c(x - y)
, 46
7.3 QUADRATIC TRINOMIALS
7.3.1 The product of (x + 2)(x + 3) = x2 + 5x + 6.
This was determined as follows:
First term: x × x = x2
Middle term: 2 × x + 3 × x = 5x
Third term: 2 × 3 = 6
The inverse process is called factorisation. The expression x2 + 5x + 6 factors can be determined
as follows:
(1) (x ) (x )
(2) (x + ) (x + )
(3) Consider the following possible factors of 6 to determine the sum of +5x for the
midterm and a product of +6 for the third term:
6=6×1 or 6=2x3
and 6x + 1x = 7x or 2x + 3x = 5x
We sum because the sign of the third term is +.
Therefore to resolve into factors x2 + 5x + 6 = (x + 2)(x + 3).
7.3.2 Similarly the product of (x + 2)(x - 4) = x2 - 2x - 8. To factorise x2 - 2x - 8 follow
the steps.
(1) (x )(x )
(2) (x + )(x - )
(3) Consider the following possible factors of 8 to determine the difference of -2x
for the midterm and a product of -8 for the third term:
8=1x8 or 8=2x4
and 1x - 8x = -7x or 2x - 4x = -2x
We subtract because the sign of the third term is -.
To resolve into factors x2 - 2x - 8 = (x + 2)(x - 4).
7.3.3 The square of a trinomial
A special trinomial expression is when both factors are the same.
The product of (x - 3)(x - 3) = x2 - 6x + 9. Look at the following properties:
(1) The coefficient of the first and third terms is a square.
(2) The coefficient of the midterm is two times the product of the square roots of the
first and third terms.
Consider: First term Third term Midterm
12 = 1 32 = 9
To resolve into factors x2 - 6x + 9 = (x - 3)2.
