12.2 - What is Factorisation

Factorisation is the process of breaking down an expression into its factors or components that can be multiplied together to give the original expression. In other words, it is the reverse of expansion, where an expression is written as the product of its factors.

In algebra, factorisation involves rewriting an algebraic expression as a product of simpler expressions. For example, a polynomial expression such as $x^2 + 5x + 6$ can be factorised into $(x + 2)(x + 3)$. The main objective of factorisation is to express an algebraic expression in its simplest form, which can make solving equations or simplifying expressions easier.

There are several methods of factorisation that are commonly used in algebra. These methods include:

• Common Factor Method
• Factorisation of Trinomial Expressions
• Difference of Squares
• Quadratic Trinomial Factorisation

Common Factor Method

In this method, we factor out the greatest common factor (GCF) from all terms in the expression. For example, consider the expression $6x + 9$. The GCF of $6x$ and $9$ is $3$. So, we can factor out $3$ to get:

$6x + 9 = 3(2x + 3)$

Here, $3$ is the common factor that has been taken out of both terms.

Factorisation of Trinomial Expressions

A trinomial expression of the form $ax^2 + bx + c$ can often be factorised into two binomial expressions. The factorisation depends on finding two numbers whose product is $ac$ (the product of the coefficient of $x^2$ and the constant term), and whose sum is $b$ (the coefficient of $x$).

For example, factorise $x^2 + 5x + 6$. Here, $a = 1$, $b = 5$, and $c = 6$. We need to find two numbers whose product is $1 \times 6 = 6$, and whose sum is $5$. These numbers are $2$ and $3$. Therefore, the factorisation is:

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

Difference of Squares

The difference of squares is a special case of factorisation. The difference of squares formula is:

$a^2 - b^2 = (a - b)(a + b)$

For example, factorise $x^2 - 16$. We recognize that $x^2$ is a square and $16$ is a square, so applying the difference of squares formula gives:

$x^2 - 16 = (x - 4)(x + 4)$

Quadratic Trinomial Factorisation

A quadratic trinomial is an expression of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. To factorise quadratic trinomials, we look for two numbers whose product is $a \times c$ and whose sum is $b$. Once these numbers are found, we split the middle term ($bx$) and factor by grouping.

For example, to factorise $2x^2 + 7x + 3$, we find two numbers whose product is $2 \times 3 = 6$ and whose sum is $7$. These numbers are $6$ and $1$. So, we rewrite the expression as:

$2x^2 + 7x + 3 = 2x^2 + 6x + x + 3$

Next, group the terms and factor:

$2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3)$

Finally, factor out the common binomial factor $(x + 3)$:

$2x^2 + 7x + 3 = (2x + 1)(x + 3)$