### Factorisation by example

#### Basic Factorisation

The process of taking common factors out in an algebraic expression is called factorising

E.g.1

2x + 8
2(x + 4)

E.g.2

2x + 8y
2(x + 4y)

E.g.3

4x2 + 6x
2x(2x + 3)

E.g.4

ax2p - 2ax3r
ax2(p - 2xr)

#### Factorising in Pairs

In this method, we pair up the terms and then factorize twice as follows:
x2 + 6x + 2x + 12
x2 + 6x + 6x + 12
First factorizing:
x(x + 6) + 2(x + 6)
Second factorizing:
(x + 6)(x + 2)

E.g.1

2ax + 6ay + bx + 3by
2a(x + 3y) + b(x + 3y)
(x + 3y)(2a + b)

E.g.2

xk - xl - yk + yl
x(k - l) - y(k - l)
(k - l)(x - y)

E.g.3

x2 - 6x + 4x - 24
x(x - 6) + 4(x - 6)
(x - 6)(x + 4)

E.g.3

x2 - 3x - 2x + 6
x(x - 3) - 2(x - 3)
(x - 3)(x - 2)

#### Factorizing Quadratic Expressions - easier

An expression with the highest term of x being a squared one, is called a quadratic expression.

E.g.1

x2 + 6x + 8
Think of two factors of 8 that add up to 6 - 4 and 2.
Now, split up the middle term into 4x and 2x
x2 + 4x + 2x + 8
Now, factorize in pairs
x2 + 4x + 2x + 8
x(x + 4) + 2(x + 4)
(x + 4)(x + 2)

E.g.2

x2 - 6x + 8
Think of two factors of 8 that add up to -6 - -4 and -2.
Now, split up the middle term into -4x and -2x
x2 - 4x - 2x + 8
Now, factorize in pairs
x2 - 4x - 2x + 8
x(x - 4) - 2(x - 4)
(x - 4)(x - 2)

E.g.3

x2 + 6x - 16
Think of two factors of -16 that add up to 6 - 8 and -2.
Now, split up the middle term into 8x and -2x
x2 + 8x - 2x - 16
Now, factorize in pairs
x2 + 8x - 2x - 16
x(x + 8) - 2(x + 8)
(x + 8)(x - 2)

#### Factorizing Quadratic Expressions - harder

E.g.1

2x2 + 13x + 6
Multiply 2 and 3 first - 2 x 6 = 12.
Think of two factors of 12 that add up to 13 - 12 and 1.
Now, split up the middle term into 12x and x
2x2 + 12x + x + 6
Now, factorize in pairs
2x2 + 12x + x + 6
2x(x + 6) + 1(x + 6)
(x + 6)(2x + 1)

E.g.2

3x2 - 11x + 6
Multiply 3 and 6 first - 3 x 6 = 18.
Think of two factors of 18 that add up to -11 - -9 and -2.
Now, split up the middle term into -9x and -2x
3x2 - 9x - 2x + 6
Now, factorize in pairs
3x2 - 9x - 2x + 6
3x(x - 3) - 2(x - 3)
(x - 3)(3x - 2)

E.g.3

4x2 - 8x - 5
Multiply 4 and 5 first - 4 x 5 = -20.
Think of two factors of 20 that add up to -8 - -10 and 2.
Now, split up the middle term into -10x and 2x
4x2 - 10x + 2x - 5
Now, factorize in pairs
4x2 - 10x + 2x - 5
2x(2x - 5) + 1(2x - 5)
(2x - 5)(2x + 1)

#### Factorizing Difference of Squares

x2 - y2 = (x + y)(x - y)

E.g.1

x2 - 9
x2 - 32
(x + 3)(x - 3)

E.g.2

4x2 - 9y2
(2x)2 - (3y)2
(2x + 3y)(2x - 3y)

E.g.3

x2 - 9/4
x2 - (3/2)2
(x + 3/2)(x - 3/2)

E.g.4

x3 - 9x/4
x[x2 - 9/4]
x[x2 - (3/2)2]
x[(x + 3/2)(x - 3/2)]
x(x +3/2)(x - 3/2)

E.g.4

Find 1012 - 992
(101 - 99)(101 + 99)
2 x 200
400

1. 3x2 - 12x
2. x2 - 12x + 20
3. 2x2 - 9x -5
4. x2 - 25/49
5. x3 - 36x/81

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