# Factorising

**H1 Factorising Monic Quadratics**

- Factorising is the process of
**expressing an algebraic expression as a product of its factors**. - A monic quadratic is one in which the
**leading coefficient (the coefficient of the highest power of x) is 1**. - To factorise a monic quadratic, we are looking for two numbers that
**multiply to give the constant term**and**add to give the coefficient of the middle term**. - For example, for the quadratic x^2 - 5x + 6, the numbers -2 and -3 both multiply to make +6 and add to make -5, so the factorization is
**(x - 2)(x - 3)**.

**H1 Factorising Non-monic Quadratics**

- Non-monic quadratics are harder to factorise as the coefficient of x^2 is not 1.
- Mutiply the coefficient of x^2 with the constant term, then find two numbers that add up to coefficient of x and multiply to give the product computed previously.
- For example, for 2x^2 +7x + 3, since (2 * 3) = 6, the numbers we are looking for are 6 and 1 because they add up to 7 and multiply to give 6. Now, rewrite the middle term (7x), splitting it into 6x + x and factorise by grouping to give the answer:
**(2x + 1)(x + 3)**.

**H1 Factorising the Difference of Two Squares**

- Understand the difference of two squares:
**(a^2 - b^2)**can be factorised to**(a - b)(a + b)**. - For example, x^2 - 9 can be expressed as (x - 3)(x + 3) because 9 is a perfect square.

**H1 Factorising the Sum and Difference of Cubes**

- The sum of cubes formula:
**a^3 + b^3 = (a + b)(a^2 - ab + b^2)**. - The difference of cubes formula:
**a^3 - b^3 = (a - b)(a^2 + ab + b^2)**. - For example, x^3 - 27 can be factorised to (x - 3)(x^2 + 3x + 9).

**H1 Common Factor**

- Always look out for
**common factors**in all terms, if present, factor them out first. - For example, in 2x^2 -12x, the common factor of 2 and x can be taken out to give
**2x(x - 6)**.

Remember, regular practise of factorising problems is key to becoming comfortable with the process. As you practise more, you will start to notice patterns, and the process will get easier.