Factorisation - Factorisation using Identities

Factorisation is the process of expressing an algebraic expression as a product of its simplest factors. Think of it as the reverse of multiplication; if you have a product, you are looking for the original 'dimensions' that created that area.

The identity $a^2 - b^2 = (a - b)(a + b)$ is used to factorise the 'Difference of Two Squares'. Visually, imagine a large square of area $a^2$ with a smaller square of area $b^2$ removed from its corner. The remaining area can be cut and rearranged into a single rectangle with dimensions $(a-b)$ and $(a+b)$.

A Perfect Square Trinomial follows the pattern $a^2 + 2ab + b^2 = (a + b)^2$. This represents the area of a large square with side length $(a+b)$. Geometrically, this large square is made of one square of area $a^2$, another square of area $b^2$, and two identical rectangles each having an area of $ab$.

The identity $a^2 - 2ab + b^2 = (a - b)^2$ is used when the middle term of a trinomial is negative. To identify this, check if the first and last terms are positive perfect squares and if the middle term is exactly twice the product of their square roots.

Always check for a Greatest Common Factor (GCF) before applying identities. For example, in $3x^2 - 27$, you should first factor out $3$ to get $3(x^2 - 9)$, which then allows you to use the difference of squares identity on the expression inside the bracket.

Identification of terms is key: Always rewrite the given terms as perfect squares before applying the identity. For instance, rewrite $16x^2$ as $(4x)^2$ and $9y^2$ as $(3y)^2$ to clearly see that $a = 4x$ and $b = 3y$.

Multi-step Factorisation: Sometimes an expression requires using identities more than once. For example, $x^4 - y^4$ is initially $(x^2)^2 - (y^2)^2 = (x^2 - y^2)(x^2 + y^2)$. The first factor $(x^2 - y^2)$ can be factorised further into $(x - y)(x + y)$.