Factorisation - Factorisation by Regrouping Terms

Regrouping is a factorisation technique used when the terms of an algebraic expression do not have a common factor across all of them, but can be arranged into smaller groups that share common factors. Visually, you look at the expression as segments, like two distinct blocks of terms.

The most common grouping for Grade 8 is the 2-2 split, where an expression with four terms is divided into two pairs. For example, in the expression $ax + ay + bx + by$, the first pair is $(ax + ay)$ and the second pair is $(bx + by)$.

A successful regrouping must result in a Common Binomial Factor. After factoring out the Highest Common Factor (HCF) from each group, the terms inside the parentheses must be identical. If you see $a(x + y) + b(x - y)$, the binomials do not match, and you must try a different grouping.

Order of terms matters: Sometimes terms are given in a scrambled order. If grouping the first two terms doesn't work, try rearranging the terms. For instance, in $x^2 - bc + ab - xc$, rearranging it to $x^2 + ab - bc - xc$ or $x^2 - xc + ab - bc$ may make the common factors more obvious.

Handling Negative Signs: If the third term in a four-term expression is negative, you must be careful when factoring out a negative sign from the second group. Visually, imagine pulling a minus sign out like a plug; this changes the signs of all terms remaining inside that set of brackets.

The final factored form is always written as the product of the common binomial factor and the sum/difference of the remaining monomial factors. It looks like this: $(\text{Common Binomial}) \times (\text{Term 1} + \text{Term 2})$.