Algebra - Factorisation

Definition of Factorisation: Factorisation is the process of expressing a given algebraic expression as a product of two or more expressions, which are called its factors. It is essentially the reverse of expansion. Visualise this as breaking down a large rectangular area into smaller rectangles whose side lengths are the factors.

Factorisation by Taking Out Common Factors: This is the most basic method where we identify the Highest Common Factor (HCF) of all the terms in the expression and write it outside a bracket. For example, in $3x^2 + 6x$, the HCF is $3x$, so the expression becomes $3x(x + 2)$. Visualise this as extracting a common 'block' from several different structures.

Factorisation by Grouping: When an expression has four or more terms and no common factor exists for all of them, we group the terms into pairs that have a common factor. For example, in $ax + ay + bx + by$, we group them as $(ax + ay) + (bx + by)$, which becomes $a(x + y) + b(x + y)$, eventually leading to $(a + b)(x + y)$.

Trinomials (Splitting the Middle Term): For a quadratic trinomial of the form $ax^2 + bx + c$, we find two numbers $p$ and $q$ such that $p + q = b$ and $p \times q = ac$. The middle term $bx$ is then replaced by $px + qx$, and factorisation is completed by grouping. Think of this as adjusting the internal segments of a rectangle to find consistent side lengths.

Difference of Two Squares: An expression in the form $a^2 - b^2$ is factorised into $(a + b)(a - b)$. Visually, this represents taking a large square of side $a$ and cutting out a smaller square of side $b$ from the corner; the remaining L-shaped area can be rearranged into a rectangle with dimensions $(a+b)$ and $(a-b)$.

Sum and Difference of Two Cubes: These are specific identities used for third-degree polynomials. The sum $a^3 + b^3$ factors into a linear part $(a+b)$ and a quadratic part $(a^2 - ab + b^2)$, while the difference $a^3 - b^3$ factors into $(a-b)(a^2 + ab + b^2)$. Notice the sign change in the middle term of the quadratic factor.

Perfect Square Trinomials: Expressions like $a^2 + 2ab + b^2$ and $a^2 - 2ab + b^2$ are perfect squares of binomials $(a+b)^2$ and $(a-b)^2$ respectively. On a grid, these expressions would form a perfect square shape where the sides are of equal length $(a+b)$ or $(a-b)$.