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.