Key Points

Factorisation is the process of writing an algebraic expression as a product of its factors. These factors can be numbers, algebraic variables, or other algebraic expressions.

An irreducible factor is a factor that cannot be expressed further as a product of factors. For example, in the expression $7xy$, the irreducible factors are $7$, $x$, and $y$.

This method involves identifying factors that are common to all terms in an expression and using the distributive law to pull them out. For example, in $3x + 9$, the common factor is $3$, so it factorises to $3(x+3)$.

When no single factor is common to all terms, group the terms so that each group has a common factor. This often reveals a common binomial factor. For example, $2xy+3x+2y+3 = x(2y+3) + 1(2y+3) = (2y+3)(x+1)$.

The identity $a^2 + 2ab + b^2 = (a+b)^2$ is used to factorise perfect square trinomials. For example, $x^2 + 6x + 9 = (x)^2 + 2(x)(3) + (3)^2 = (x+3)^2$.

The identity $a^2 - 2ab + b^2 = (a-b)^2$ is used for factorisation. For example, $p^2 - 10p + 25 = (p)^2 - 2(p)(5) + (5)^2 = (p-5)^2$.

The identity $a^2 - b^2 = (a+b)(a-b)$ is used to factorise an expression that is the difference of two perfect squares. For example, $4x^2 - 9y^2 = (2x)^2 - (3y)^2 = (2x+3y)(2x-3y)$.

To factorise $x^2 + px + q$, find two numbers, $a$ and $b$, such that their sum $a+b=p$ and their product $ab=q$. The factorised form is $(x+a)(x+b)$. For $y^2 - 7y + 12$, the numbers are $-3$ and $-4$, so the factors are $(y-3)(y-4)$.

To divide a monomial by another, express each as a product of irreducible factors and cancel the common factors. For example, $28x^4 \div 56x = \frac{28 \times x \times x \times x \times x}{56 \times x} = \frac{1}{2}x^3$.

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial individually. For example, $(5x^2 - 6x) \div 3x = \frac{5x^2}{3x} - \frac{6x}{3x} = \frac{5}{3}x - 2$.

To divide a polynomial by another polynomial, factorise both the dividend (numerator) and the divisor (denominator), and then cancel any common factors. For example, $(y^2 + 7y + 10) \div (y+5) = \frac{(y+2)(y+5)}{(y+5)} = y+2$.

The fundamental relationship in division is Dividend = (Divisor $\times$ Quotient) + Remainder. The methods in this chapter apply to cases where the remainder is zero.