Factorisation - Factorisation by Regrouping Terms

Factorisation is the process of rewriting an algebraic expression as the product of its constituent factors. For example, the expression $2x + 4$ is factorised into $2(x + 2)$, where $2$ and $(x + 2)$ are factors.

Regrouping terms is a technique used when all terms in an expression do not share a single common factor, but subsets of terms do. It involves rearranging and clustering terms into groups (usually pairs) to facilitate factorisation.

Visualizing the Grouping: When looking at a four-term expression like $ax + ay + bx + by$, imagine it as two distinct blocks: $(ax + ay)$ and $(bx + by)$. This mental separation helps identify the common factor within each 'block' independently.

Application of the Distributive Law in Reverse: Regrouping relies on the identity $a(b + c) = ab + ac$. By reversing this, we transform the sum of terms $ab + ac$ back into the product $a(b + c)$.

Identifying the 'Twin Parentheses': After the first stage of grouping, a successful regrouping will result in two terms sharing an identical binomial factor, such as $p(x + y) + q(x + y)$. Visualising these identical brackets as a single common entity allows you to pull them out as a single factor.

Managing Signs and Negative Factors: Careful attention must be paid to signs when grouping. For example, in the expression $ax - ay - bx + by$, when you group the last two terms as $-(bx - by)$, the signs inside the bracket 'flip' to ensure the binomial factor $(x - y)$ matches the first group.

Final Product Verification: The end result of factorisation by regrouping should always be a product of two or more algebraic expressions. You can verify the result by multiplying the factors back together to see if you reach the original expression.