Factorisation - Method of Common Factors

Factorisation is the process of expressing an algebraic expression as a product of two or more expressions, called its factors. Visually, if an expression represents the area of a rectangle, factorisation is the process of finding the lengths of its sides.

The Method of Common Factors involves identifying the Highest Common Factor (HCF) of all the terms in an expression and then dividing each term by this HCF. Think of this as the 'reverse distributive law' where you move from $ab + ac$ to $a(b + c)$.

To find the HCF of monomials, find the HCF of the numerical coefficients and multiply it by each common variable raised to its lowest power. For example, in the terms $8x^{3}y^{2}$ and $12x^{2}y^{4}$, the numerical HCF is $4$, and the variable HCF is $x^{2}y^{2}$, making the total HCF $4x^{2}y^{2}$.

When factorising by grouping, terms of an expression are arranged into groups such that each group has a common factor. This is often used when there is no single factor common to all terms. For example, in $ax + ay + bx + by$, the first two terms share $a$ and the last two share $b$.

The sign of the common factor is crucial. If you extract a negative common factor, you must change the signs of all terms inside the brackets. For instance, $-5x - 10$ becomes $-5(x + 2)$. Visually, this ensures that if you were to expand the brackets again, you would return to the original expression.

A common binomial factor can also be extracted. If an expression looks like $x(a + b) + y(a + b)$, the entire bracket $(a + b)$ is treated as a single common factor, resulting in $(a + b)(x + y)$. This looks like two rectangular blocks sharing a common side length $(a + b)$ placed side by side.