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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 of Algebraic Expressions - Continued

This section continues the study of factorisation by introducing more complex algebraic expressions and their factorisation. It covers factorisation of cubic expressions and the use of identities such as the sum and difference of cubes. The section explains how expressions like a³ + b³ can be factorised as (a + b)(a² - ab + b²) and a³ - b³ as (a - b)(a² + ab + b²). These identities are useful for factorising higher-degree polynomials. The section also revisits factorisation by grouping for expressions with more than four terms and demonstrates the use of substitution to simplify factorisation. Several examples illustrate these methods, emphasizing careful identification of patterns and correct application of formulas. The section concludes by highlighting the importance of practice in mastering these techniques.

📊 Diagram: Diagrams illustrating factorisation of sum and difference of cubes, showing the breakdown into binomial and trinomial factors.

🧪 Activity: Exercises to factorise cubic expressions using sum and difference of cubes formulas and grouping.

🔗 Connection: Leads to the next section 'Factorisation and Algebraic Identities' which connects factorisation with algebraic identities.

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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