This is the process of expressing an algebraic expression as a product of its factors e.g.

- $ 5x^2 + 3x = x(5x + 3)$
- $ x^2 +5x +6x^4 = x(x + 5 + 6x^3)$

The degree of a polynomial is the value of the highest exponent e.g.

$x^2 + 5x+6 $ is a quadratic (highest power is 2), trinomial (3 terms) expression.

Now for some examples of factorisation:

${x^2} +5x+6$

Step 1: Multiply 1 and 6 i.e. $ {1} \times 6 = {6}$

Step 2: Find factors of 6 i.e. $ 1, {-1}, {-2}, {-3},{-6}, 2,3, 6$

Step 3: Find factors of 6 with a product of 6 i.e. $ 2 \times 3$

Step 4:Find factors of 6 with a sum of 5 i.e. $ 2 + 3 =5 $

Step 5: Write 5 as $2x +3x $ and substitute $5x$ in the original expression thus:

${x^2} +5x+6$ becomes ${x^2}+2x +3x+6$

$= x(x+2)+3(x+2)$

$=(x+3)(x+2)$

$x^2+6x+9$

$=x^2+3x+3x+9$

$=x(x+3) +3(x+3)$

$=(x+3)(x+3)$

$2x^2-10x-12$

$=2(x^2-5x-6)$

$= 2(x-6)(x+10)$

$3x^2 – 5x-8$

$=3x^2-8x+3x-8$

$x(3x-8) + 1(3x-8) $

$(3x-8)(x-1)$

$16s^2 -8s+1$

$16s^2+4s+4s+1$

$=(4s+1)(4s+1)$

=$4s+1)^2$

**Exercise**

Factorise the following:

$x^2+x-6$

$p^2x +4pqx+4q^2x$

$a^2 -4ab+4b^2-16x^2$

$9x^2-3x+24x^3-8x^2$

**Difference of two Squares**

$x^2 – y^2 = (x+y)(x-y)$

Example 1: $ 9 – 4 = 3^2 – 2^2 = (3+2)(3-2) = 5$

Example 2: $ 16x^2 – 9y^2 = 4^2x^2 -3^2y^2 = (4x +3y)(4x-3y) $

Example 3: $ 7x^2 – 28b^2 = 7(x^2-4b^2)= 7(x+2b)(x-2b) $

Example 4: $ x^8 – y^6 = (x^4)^2 – (y^3)^2 = (x^4 -y^3)(x^4+y^3)$

**Difference of two Cubes**

$x^3-y^3 = (x-y)(x^2+xy+y^2)$

Example 1: $27 – 8 = 3^3 – 2^3 = (3-2)(3^2+3(2)+2^2) = 1(9+6+4) = 19 $

Example 2: $ 8x^3 -27 = 2^3x^3 – 3^3 = (2x-3)((2x)^2+2x(3)+3^2) = (2x-3)(4x^2+6x+9)$

**Sum of two Cubes**

$x^3+y^3 = (x+y)(x^2-xy+y^2)$

Example 1: $ 27 + 8 = 3^3 + 2^3 = (3+2)(3^2-3(2)+2^2) = 5(9-6+4) = 35 $

Example 2: $ 8x^3 + 27 = 2^3x^3 + 3^3 = (2x+3)((2x)^2-2x(3)+3^2) = (2x+3)(4x^2-6x+9)$

**Sum of two Squares**

$x^2 + y^2 = x^2 + y^2 $

Example 1: $ 9 + 4 = 3^2 + 2^2 = 9 + 4 = 13 $

Example 2: $ 16x^2 + 9y^2 = 4^2x^2 -3^2y^2 = 16x^2 + 9y^2 $

**Exercise**

- $x^2-1$
- $(x-y)^2-4$
- $x^2+9xy+8y$
- $x^2-11x+24$

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