Factorisation basically means to take out the common factors in a given equation.

Example:
Factorise the following equation
x² + 2x+ x³

The common factor in the above equation is ‘x’. So we get:
x (x + 2 + x²)

If we open the brackets, we get the original equation.
x (x + 2 + x²) = x² + 2x + x³

Example:
Factorise the following equation
x³ + x²y + 2x²

Here, the common factor is x². So we get:
x² (x + y + 2)

Example:
Factorise the following equation
2x⁴ + 4x²y + 12x³z

Here, the common factor is 2x². We get:`
2x² (x² + 2y + 6xz)

### Splitting the Middle Term

It is useful to know how to factorise quadratic equations by splitting the middle term. Here is how we do it:

You are given the quadratic equation:
2x² + 11x + 12

Step 1: You multiply the third term (+12) with the first term (2x²). We get:
2x² * 12 = 24x²

Step 2: We must split the middle term (11x) into two parts in such a way that the product of the two parts would be 24x² and their sum would be 11x.
The two parts would be: 8x and 3x.
Product of two parts = 8x * 3x = 24x²
Sum of two parts = 8x + 3x = 11x

Step 3: We get the following equation:
2x² + 8x + 3x + 12

Step 4: We factorise the first two terms and the last two terms. We get:
2x (x + 4) + 3 (x + 4)

Step 5: We get a common expression (x + 4) in the equation and the other expression is (2x + 3)

Step 6: We have factorised the equation!
(2x + 3) (x + 4)

Try multiplying the brackets and you will see that:
(2x + 3) (x + 4) = 2x² + 11x + 12