Factorisation - Factorisation using Identities

Factorisation is the process of writing an algebraic expression as a product of its factors. Using identities is a shortcut method where we recognize the pattern of the expression and match it to a known algebraic formula to find its factors directly.

Identity I: The Square of a Sum. This identity is used for perfect square trinomials where all terms are positive. Visually, this corresponds to a large square with side length $(a + b)$, which is composed of a smaller square of area $a^2$, two rectangles each of area $ab$, and another small square of area $b^2$. The expression $a^2 + 2ab + b^2$ factors into $(a + b)(a + b)$ or $(a + b)^2$.

Identity II: The Square of a Difference. This identity applies to trinomials where the middle term is negative. It follows the pattern $a^2 - 2ab + b^2$. Visually, it represents the area of a square with side $(a - b)$ obtained by subtracting segments of length $b$ from a larger square of side $a$. This factors into $(a - b)(a - b)$ or $(a - b)^2$.

Identity III: Difference of Two Squares. This identity is used when an expression consists of two perfect square terms separated by a minus sign ($a^2 - b^2$). Visually, imagine taking a square of side $a$ and cutting out a smaller square of side $b$ from one corner. The remaining L-shaped area can be sliced and rearranged to form a rectangle with sides $(a + b)$ and $(a - b)$. The factors are $(a + b)(a - b)$.

Identity IV: Product of Binomials with a Common Term. This is used for trinomials of the form $x^2 + (a + b)x + ab$. Here, we look for two numbers whose sum is the coefficient of $x$ and whose product is the constant term. Visually, this represents a rectangle with dimensions $(x + a)$ and $(x + b)$ made up of a square $x^2$, two rectangles $ax$ and $bx$, and a small rectangle $ab$. The factors are $(x + a)(x + b)$.

Recognition Strategy: To factorise effectively, first check if there is a common factor among all terms. Then, count the terms. If there are two terms, check for the Difference of Squares. If there are three terms, check if the first and last terms are perfect squares to apply Identity I or II; otherwise, try Identity IV.

Verification: After factorising, you can always verify your answer by multiplying the factors back together using the distributive property. The resulting expansion must equal the original algebraic expression.