Class 8 Maths Chapter 14 Factorisation Of Quadratic Trinomials Part 1 - Factorisation

Factorisation: Algebraic Factorisation of Quadratic Trinomials and Identities

Factorisation is the process of breaking down an algebraic expression into a product of simpler expressions. Just as a number like 30 can be expressed as the product of its factors (2 × 3 × 5), algebraic expressions can be factorised to reveal their underlying structure. Factorisation is an essential skill for solving quadratic equations, simplifying rational expressions, and understanding the behaviour of polynomial functions. This chapter in CBSE Class 8 Mathematics introduces methods of factorisation including common factors, grouping, the difference of squares, and the factorisation of quadratic trinomials using splitting the middle term.

The simplest method of factorisation is finding common factors. If every term in an expression shares a common factor (numerical, variable, or both), we use the distributive law in reverse: ab + ac = a(b + c). For example, 6x² + 9x = 3x(2x + 3). The rule for division of monomials or polynomials by a monomial: if we have an expression of the form a²b + ab², common factor ab gives ab(a + b). The method of grouping is used when an expression has four or more terms and no single factor is common to all terms. In grouping, we group terms that share common factors, factor each group separately, and then look for a common binomial factor. For example, x³ − x² + 3x − 3 = x²(x − 1) + 3(x − 1) = (x − 1)(x² + 3). The three standard algebraic identities are used both for expanding and factoring expressions: (x + a)(x + b) = x² + (a + b)x + ab; (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; and a² − b² = (a + b)(a − b). For example, 49x² − 64y² = (7x)² − (8y)² = (7x + 8y)(7x − 8y).

The splitting (splitting the middle term) method is used to factorise quadratic trinomials of the form ax² + bx + c. The steps are: (1) Multiply the coefficient of x² (a) by the constant term (c) to get ac. (2) Find two numbers p and q whose product equals ac and whose sum equals b (the coefficient of x). (3) Split the middle term into these two parts: ax² + px + qx + c. (4) Factor by grouping. For example, to factorise x² + 7x + 12: ac = 1×12 = 12, and we need two numbers whose product is 12 and sum is 7. The numbers are 3 and 4 (3×4=12, 3+4=7). Splitting: x² + 3x + 4x + 12 = x(x+3) + 4(x+3) = (x+3)(x+4). For 2x² + 7x + 3: ac = 2×3 = 6, numbers 6 and 1 (6×1=6, 6+1=7): 2x² + 6x + 1x + 3 = 2x(x+3) + 1(x+3) = (x+3)(2x+1). After factorisation, we can verify our answer by expanding the factors — the product should equal the original expression. Division of polynomials can be performed using long division, but factorisation often provides a faster method: if we know the factors, the quotient is obtained by dividing by one factor. The division of an algebraic expression by a monomial is done by dividing each term of the expression by the monomial separately. For example, (6x³ + 9x² − 3x) ÷ 3x = 6x³/3x + 9x²/3x − 3x/3x = 2x² + 3x − 1.

• Factorisation by common factors: ab + ac = a(b + c); grouping: pair terms to find common binomial factors.
• Algebraic identities: (a+b)² = a²+2ab+b², (a-b)² = a²−2ab+b², a²−b² = (a+b)(a−b).
• Splitting the middle term for ax²+bx+c: multiply a and c, find factors whose sum is b, split and group.
• Check factorisation by expanding the factors — the product must equal the original expression.
• Division by a monomial: divide each term separately; use factorisation to simplify polynomial division.