Summary
Chapter 12 of the Class 8 Maths NCERT textbook, "Factorisation", teaches students how to express algebraic expressions as products of their factors. The chapter covers systematic methods for finding factors of both simple and complex algebraic expressions, including factorisation by common factors, regrouping, identities, and division of algebraic expressions.
- Factorisation as reversal — Factorising rewrites an expression as a product of irreducible factors — the algebraic counterpart of prime factorisation — and the chapter offers a toolkit of methods for doing so systematically.
- Common factors and regrouping — The first tools pull out the greatest common factor via the distributive law, and, when no single common factor exists, regroup terms strategically to expose a shared factor across the groups.
- Identities and division — Recognising expressions as squares of sums/differences or differences of squares lets identities be applied in reverse, while factorisation also powers division by cancelling common factors.
Key points & formulas
- 01An irreducible factor is a factor that cannot be expressed further as a product of other factors; prime factorization and algebraic factorization both identify irreducible units.
- 02Common factor method: extract the greatest common factor from all terms, then apply the distributive law (e.g., 5xy + 10x = 5x(y + 2)).
- 03Regrouping method: when terms lack a single common factor, group them strategically to reveal a common factor across groups (e.g., 2xy + 2y + 3x + 3 = (x + 1)(2y + 3)).
- 04Factorisation using identities: recognize patterns matching (a + b)² = a² + 2ab + b², (a – b)² = a² – 2ab + b², or (a + b)(a – b) = a² – b², then apply the identity in reverse.
- 05Quadratic factorisation: for x² + px + q, find two factors a and b of q such that ab = q and a + b = p, giving (x + a)(x + b).
- 06Division as inverse of multiplication: factorise both dividend and divisor, then cancel common factors (e.g., (7x² + 14x) ÷ (x + 2) = 7x(x + 2) ÷ (x + 2) = 7x).
- 07Multiple identity application: combine two or more identities to factorise complex expressions (e.g., a² – 2ab + b² – c² = (a – b)² – c² = (a – b – c)(a – b + c)).
Frequently asked questions
01What is factorisation in algebra?
Factorisation is the process of writing an algebraic expression as a product of its factors. Factors may be numbers (like 2 or 5), algebraic variables (like x or y), or algebraic expressions (like (x + 2)). For example, 2x + 4 factorises to 2(x + 2), where 2 and (x + 2) are the factors.
02What is the common factor method?
The common factor method involves three steps: (1) Write each term of the expression as a product of irreducible factors, (2) identify the factors that appear in every term (the common factors), and (3) extract the common factors and combine the remaining factors using the distributive law. For example, in 12a²b + 15ab², the common factors are 3, a, and b, so the result is 3ab(4a + 5b).
03When do I use regrouping to factorise an expression?
Use regrouping when no single factor is common to all terms in the expression. Group the terms strategically so that each group has a common factor, then extract that common factor from each group. A new common factor often emerges across the groups, allowing full factorisation. For example, 2xy + 2y + 3x + 3 groups as (2xy + 2y) + (3x + 3) = 2y(x + 1) + 3(x + 1) = (x + 1)(2y + 3).
04What algebraic identities help with factorisation?
Four key identities are used: (a + b)² = a² + 2ab + b², (a – b)² = a² – 2ab + b², (a + b)(a – b) = a² – b², and (x + a)(x + b) = x² + (a + b)x + ab. To factorise, recognise which identity pattern the expression matches and apply it in reverse. For example, x² + 8x + 16 matches a² + 2ab + b² with a = x and b = 4, so it factorises to (x + 4)².
05How do I factorise x² + 5x + 6?
For x² + px + q, find two numbers a and b such that ab = 6 (the constant term) and a + b = 5 (the coefficient of x). Here, a = 2 and b = 3 work (2 × 3 = 6 and 2 + 3 = 5). Rewrite as x² + 2x + 3x + 6, factor by grouping to get x(x + 2) + 3(x + 2) = (x + 2)(x + 3). You can verify: (x + 2)(x + 3) = x² + 5x + 6.
More chapters in Mathematics
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