Factorisation - Factorisation of the form (x+a)(x+b)

The identity $(x+a)(x+b) = x^2 + (a+b)x + ab$ is the fundamental basis for factorising quadratic trinomials where the coefficient of $x^2$ is $1$. Factorisation is essentially the reverse process of expanding these brackets.

A quadratic trinomial in the form $x^2 + px + q$ can be factorised into $(x+a)(x+b)$ by finding two integers $a$ and $b$ such that their sum $a + b = p$ and their product $ab = q$.

The 'Sign Rule' for the constant term: If the constant term $q$ is positive, both $a$ and $b$ must have the same sign as the middle term $p$. If $q$ is negative, $a$ and $b$ must have opposite signs, and the larger factor (in absolute value) will carry the same sign as the middle term $p$.

Splitting the Middle Term: Once $a$ and $b$ are identified, the term $px$ is replaced with $ax + bx$. This transforms the trinomial into a four-term expression $x^2 + ax + bx + ab$, which can then be factorised by grouping terms into pairs.

Visualizing with an Area Model: Imagine a large rectangle with dimensions $(x+a)$ and $(x+b)$. The total area of this rectangle represents the trinomial $x^2 + (a+b)x + ab$. Inside this large rectangle, you can see four distinct regions: a square of area $x^2$, two rectangles of areas $ax$ and $bx$, and a smaller corner rectangle of area $ab$.

Systematic Factor Search: To find $a$ and $b$, list all factor pairs of the constant term $q$. For each pair, calculate their sum until you find the pair that equals the coefficient $p$. For example, if $q = 12$, pairs include $(1, 12), (2, 6),$ and $(3, 4)$.