Factorisation - Method of Common Factors

Factorisation is the process of writing an algebraic expression as a product of two or more expressions. It is the reverse of the Distributive Law. Visualise this like taking a complex machine and dismantling it into its original component parts or 'factors'.

Factors of Monomials: A monomial can be broken down into its irreducible factors. For example, $5xy$ has factors $5, x,$ and $y$. These are called 'irreducible' because they cannot be factorised further, much like prime numbers in arithmetic.

Method of Common Factors: This involves identifying the Highest Common Factor (HCF) of all the terms in the expression. Think of this as looking at a row of boxes and picking out the exact same item that appears in every single box to place it outside.

The Distributive Property in Reverse: In multiplication, we use $a(b + c) = ab + ac$. In factorisation, we observe the common $a$ in $ab + ac$ and pull it out to get $a(b + c)$. This can be visualised as finding the common width 'a' for two adjacent rectangles with areas $ab$ and $ac$.

Identifying Numerical HCF: For the numerical coefficients, find the largest number that divides all of them. For example, in $12x + 18$, the HCF of $12$ and $18$ is $6$. On a number line, this is the largest step size that lands exactly on both $12$ and $18$.

Identifying Literal (Variable) HCF: For the variables, look for the lowest power of each common variable. In $x^{3}$ and $x^{2}$, the common factor is $x^{2}$ because it is the largest power contained within both terms.

Irreducible Form: An expression is said to be in its factorised form when no further common factors can be extracted. Visually, this is represented as a product of brackets where the contents inside each bracket have no common factors remaining.