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MathematicsClass 8Factorisation

Factorisation | Class 8 Mathematics Notes

By ConceptScroll Team · Published on 17 July 2026 · 5 min read

Factorisation – this guide gives you a concise, exam-ready overview of Factorisation from Class 8 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.

Factorisation and Algebraic Identities

This section establishes the connection between factorisation and algebraic identities. Algebraic identities are equations that hold true for all values of the variables involved. The chapter revisits important identities such as (a + b)² = a² + 2ab + b², (a - b)² = a² - 2ab + b², a² - b² = (a - b)(a + b), and the sum and difference of cubes identities. These identities are not only useful for expansion but also for factorisation. Recognizing these patterns in algebraic expressions allows for quick and efficient factorisation. The section provides examples where expressions are factorised by applying these identities in reverse. This understanding is crucial for simplifying expressions, solving equations, and performing algebraic manipulations. The section also encourages students to memorize these identities for ease of use.

📊 Diagram: Diagrams illustrating the expansion and factorisation of algebraic identities, showing the equivalence between expressions and their factored forms.

🧪 Activity: Exercises to factorise expressions by identifying and applying algebraic identities.

🔗 Connection: Prepares for the next section 'Factorisation of Polynomials' which applies these concepts to polynomials.

Frequently asked questions

1. Find the common factors of the given terms. (i) 12x, 36 (ii) 2y, 22xy (iii) 14 pq, 28 p^2 q^2 (iv) 2x, 3x^2, 4 (v) 6 abc, 24 ab^2, 12 a^2 b (vi) 16 x^3, -4 x^2, 32 x (vii) 10 pq, 20 qr, 30 rp (viii) 3 x^2 y^3, 10 x^3 y^2, 6 x^2 y^2 z

Solutions:

(i) 12x, 36 Common factors: Factors of 12x are 1,2,3,4,6,12,x; Factors of 36 are 1,2,3,4,6,9,12,18,36. Common factors: 1,2,3,4,6,12

(ii) 2y, 22xy Factors of 2y: 1,2,y Factors of 22xy: 1,2,11,x,y,22, etc. Common factors: 1,2,y

(iii) 14 pq, 28 p^2 q^2 Factors of 14 pq: 1,2,7,p,q Factors of 28 p^2 q^2: 1,2,4,7,14,p,p^2,q,q^2 Common factors: 1,2,7,p,q

(iv) 2x, 3x^2, 4 Factors of 2x: 1,2,x Factors of 3x^2: 1,3,x,x^2 Factors of 4: 1,2,4 Common factors: 1

(v) 6 abc, 24 ab^2, 12 a^2 b Fa

2. Factorise the following expressions. (i) 7x - 42 (ii) 6p - 12q (iii) 7a^2 + 14a (iv) -16z + 20z^3 (v) 20 l^2 m + 30 a l m (vi) 5 x^2 y - 15 x y^2 (vii) 10 a^2 - 15 b^2 + 20 c^2 (viii) -4 a^2 + 4 a b - 4 c a (ix) x^2 y z + x y^2 z + x y z^2 (x) a x^2 y + b x y^2 + c x y z

Solutions:

(i) 7x - 42 = 7(x - 6)

(ii) 6p - 12q = 6(p - 2q)

(iii) 7a^2 + 14a = 7a(a + 2)

(iv) -16z + 20z^3 = 4z(-4 + 5z^2) = 4z(5z^2 - 4)

(v) 20 l^2 m + 30 a l m = 10 l m (2 l + 3 a)

(vi) 5 x^2 y - 15 x y^2 = 5 x y (x - 3 y)

(vii) 10 a^2 - 15 b^2 + 20 c^2 = 5 (2 a^2 - 3 b^2 + 4 c^2)

(viii) -4 a^2 + 4 a b - 4 c a = -4 a^2 + 4 a b - 4 a c = -4 a (a - b + c)

(ix) x^2 y z + x y^2 z + x y z^2 = x y z (x + y + z)

(x) a x^2 y + b x y^2 + c x y z = x y (a x + b y + c z)

3. Factorise. (i) x^2 + x y + 8 x + 8 y (ii) 15 x y - 6 x + 5 y - 2 (iii) a x + b x - a y - b y (iv) 15 p q + 15 + 9 q + 25 p (v) z - 7 + 7 x y - x y z

Solutions:

(i) x^2 + x y + 8 x + 8 y = x(x + y) + 8(x + y) = (x + 8)(x + y)

(ii) 15 x y - 6 x + 5 y - 2 = 3 x (5 y - 2) + 1 (5 y - 2) = (3 x + 1)(5 y - 2)

(iii) a x + b x - a y - b y = x(a + b) - y(a + b) = (x - y)(a + b)

(iv) 15 p q + 15 + 9 q + 25 p = 15(p q + 1) + 9 q + 25 p = 15 p q + 15 + 9 q + 25 p Try grouping: = (15 p q + 9 q) + (25 p + 15) = 3 q (5 p + 3) + 5 (5 p + 3) = (3 q + 5)(5 p + 3)

(v) z - 7 + 7 x y - x y z = (z - x y z) + (7 x y - 7) = z(1 - x y) + 7(x y - 1) = z(1 - x y)

1. Factorise the following expressions. (i) a^2 + 8a + 16 (ii) p^2 - 10p + 25 (iii) 25m^2 + 30m + 9 (iv) 49y^2 + 84yz + 36z^2 (v) 4x^2 - 8x + 4 (vi) 121b^2 - 88bc + 16c^2 (vii) (l + m)^2 - 4lm (Hint: Expand (l + m)^2 first) (viii) a^4 + 2a^2b^2 + b^4

Solutions:

(i) a^2 + 8a + 16 = (a + 4)^2

(ii) p^2 - 10p + 25 = (p - 5)^2

(iii) 25m^2 + 30m + 9 = (5m + 3)^2

(iv) 49y^2 + 84yz + 36z^2 = (7y + 6z)^2

(v) 4x^2 - 8x + 4 = (2x - 2)^2

(vi) 121b^2 - 88bc + 16c^2 = (11b - 4c)^2

(vii) (l + m)^2 - 4lm = (l + m)^2 - (2√lm)^2 = (l + m - 2√lm)(l + m + 2√lm) But since 4lm = (2l)(2m), better to write: (l + m)^2 - 4lm = (l - m)^2

(viii) a^4 + 2a^2b^2 + b^4 = (a^2 + b^2)^2

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