### 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**

4x^{2} + 6x

2x(2x + 3)

**E.g.4**

ax^{2}p - 2ax^{3}r

ax^{2}(p - 2xr)

#### Factorising in Pairs

In this method, we pair up the terms and then factorize twice as follows:

x^{2} + 6x + 2x + 12

x^{2} + 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**

x^{2} - 6x + 4x - 24

x(x - 6) + 4(x - 6)

(x - 6)(x + 4)

**E.g.3**

x^{2} - 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**

x^{2} + 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

x^{2} + 4x + 2x + 8

Now, factorize in pairs

x^{2} + 4x + 2x + 8

x(x + 4) + 2(x + 4)

(x + 4)(x + 2)

**E.g.2**

x^{2} - 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

x^{2} - 4x - 2x + 8

Now, factorize in pairs

x^{2} - 4x - 2x + 8

x(x - 4) - 2(x - 4)

(x - 4)(x - 2)

**E.g.3**

x^{2} + 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

x^{2} + 8x - 2x - 16

Now, factorize in pairs

x^{2} + 8x - 2x - 16

x(x + 8) - 2(x + 8)

(x + 8)(x - 2)

#### Factorizing Quadratic Expressions - harder

**E.g.1**

2x^{2} + 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

2x^{2} + 12x + x + 6

Now, factorize in pairs

2x^{2} + 12x + x + 6

2x(x + 6) + 1(x + 6)

(x + 6)(2x + 1)

**E.g.2**

3x^{2} - 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

3x^{2} - 9x - 2x + 6

Now, factorize in pairs

3x^{2} - 9x - 2x + 6

3x(x - 3) - 2(x - 3)

(x - 3)(3x - 2)

**E.g.3**

4x^{2} - 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

4x^{2} - 10x + 2x - 5

Now, factorize in pairs

4x^{2} - 10x + 2x - 5

2x(2x - 5) + 1(2x - 5)

(2x - 5)(2x + 1)

#### Factorizing Difference of Squares

x^{2} - y^{2} = (x + y)(x - y)

**E.g.1**

x^{2} - 9

x^{2} - 3^{2}

(x + 3)(x - 3)

**E.g.2**

4x^{2} - 9y^{2}

(2x)^{2} - (3y)^{2}

(2x + 3y)(2x - 3y)

**E.g.3**

x^{2} - 9/4

x^{2} - (3/2)^{2}

(x + 3/2)(x - 3/2)

**E.g.4**

x^{3} - 9x/4

x[x^{2} - 9/4]

x[x^{2} - (3/2)^{2}]

x[(x + 3/2)(x - 3/2)]

x(x +3/2)(x - 3/2)

**E.g.4**

Find 101^{2} - 99^{2}

(101 - 99)(101 + 99)

2 x 200

400

**Now, please practise the following:**

- 3x
^{2}- 12x - x
^{2}- 12x + 20 - 2x
^{2}- 9x -5 - x
^{2}- 25/49 - x
^{3}- 36x/81 - x
^{3}- x^{2}+ 2x - 2 - x
^{3}- y^{2}- x^{2}+ xy^{2} - 5x
^{3}+ 5x^{2}- x - 1 - ax + bx + ay + by - cx - cy
- 12x
^{2}- 8x^{3}+ 9 - 6x

`Challenge`

The sum of the two factors of **x/y - y/x** is **x/y + y/x** . Find the two factors.