In algebra, factors are the building blocks that combine tigether to form expressions. For example, in the term 5xy, the factors are 5, x, and y. We express this as 5xy = 5 * x * y. These factors cannot be broken down further into other factors.
Taking another example, the expression 4x(x + 3) is the product of the irreducible factors 4, x, and (x + 3), written as 4x(x + 3) = 4 * x * (x + 3).
What is Factorisation?
Factorization is the process of breaking down an algebraic expression into its factors. For example, expressions such as 7xz, 9x²z, and 6x(z + 4) are already factorized. Their factors are straightforward.
However, expressions like 2x + 6 or x² + 7x + 12 are not factorized. To factorize them, we must use methods to find their irreducible factors. For 2x + 6, we can factor out a 2, giving us 2(x + 3). For x² + 7x + 12, we find factors (x + 3)(x + 4) that multiply to give the original expression. These methods are important to simplifying algebraic expressions.
NCERT Solutions for Class 8 Maths Exercise 12.2 Chapter 12 Factorisation
1. Find the common factors of the given terms.
(i) 12x, 36 (ii) 2y, 22xy (iii) 14 pq, 28p²q² (iv) 2x, 3x², 4 (v) 6 abc, 24ab², 12 a²b (vi) 16 x³, – 4x², 32x (vii) 10 pq, 20qr, 30rp (viii) 3x² y³, 10x³ y²,6 x² y²z
Finding Common Factors of Given Terms:
(i) Common factors of 12x, 36
Step 1: Identify the prime factors.
12x = 2 * 2 * 3 * x
36 = 2 * 2 * 3 * 3
Step 2: Find common prime factors.
Common factors = 2 * 2 * 3 = 12
(ii) Common factors of 2y, 22xy
Step 1: Identify the prime factors.
2y = 2 * y
22xy = 2 * 11 * x * y
Step 2: Find common prime factors.
Common factors = 2 * y = 2y
(iii) Common factors of 14pq, 28p²q²
Step 1: Identify the prime factors.
14pq = 2 * 7 * p * q
28p²q² = 2 * 2 * 7 * p * p * q * q
Step 2: Find common prime factors.
Common factors = 2 * 7 * p * q = 14pq
(iv) Common factors of 2x, 3x², 4
Step 1: Identify the prime factors.
2x = 2 * x
3x² = 3 * x * x
4 = 2 * 2
Step 2: Find common prime factors.
Common factors = None (No common factors among all three terms)
(v) Common factors of 6abc, 24ab², 12a²b
Step 1: Identify the prime factors.
6abc = 2 * 3 * a * b * c
24ab² = 2 * 2 * 2 * 3 * a * b * b
12a²b = 2 * 2 * 3 * a * a * b
Step 2: Find common prime factors.
Common factors = 2 * 3 * a * b = 6ab
(vi) Common factors of 16 x³, –4x², 32x
Step 1: Identify the prime factors.
16 x³ = 2 * 2 * 2 * 2 * x * x * x
-4x² = 2 * 2 * x * x
32x = 2 * 2 * 2 * 2 * 2 * x
Step 2: Find common prime factors.
Common factors = 2 * 2 * x = 4x
(vii) Common factors of 10pq, 20qr, 30rp
Step 1: Identify the prime factors.
10pq = 2 * 5 * p * q
20qr = 2 * 2 * 5 * q * r
30rp = 2 * 3 * 5 * r * p
Step 2: Find common prime factors.
Common factors = 5 (Only 5 is common among all)
(viii) Common factors of 3x²y³, 10x³y², 6x²y²z
Step 1: Identify the prime factors.
3x²y³ = 3 * x * x * y * y * y
10x³y² = 2 * 5 * x * x * x * y * y
6x²y²z = 2 * 3 * x * x * y * y * z
Step 2: Find common prime factors.
Common factors = x * x * y * y = x²y²
2. Factorise the following expressions.
(i) 7x – 42 (ii) 6p – 12q (iii) 7a² + 14a (iv) – 16 z + 20 z³ (v) 20 l² m + 30 a l m (vi) 5 x² y – 15 xy² (vii) 10 a² – 15 b² + 20 c² (viii) – 4 a² + 4 ab – 4 ca (ix) x² y z + x y²z + x y z² (x) a x² y + b x y² + c x y z
Factorising the Following Expressions:
(i) 7x – 42
Here 7x = 7 * x and 42 = 2 * 3 * 7
Factorise:
7 * x – 2 * 3 * 7
= 7 * (x – 2 * 3)
= 7(x – 6)
(ii) 6p – 12q
Factorise:
6 * p – 6 * 2 * q
= 6(p – 2q)
(iii) 7a² + 14a
Factorise:
7 * a * a + 7 * 2 * a
= 7a(a + 2)
(iv) – 16z + 20z³
Factorise:
-4 * 4 * z + 4 * 5 * z * z * z
= 4z(-4 + 5z²)
(v) 20l²m + 30alm
Factorise:
10 * 2 * l * l * m + 10 * 3 * a * l * m
= 10 * l * m * (2l + 3a)
(vi) 5x²y – 15xy²
Factorise:
5 * x * x * y – 5 * 3 * x * y * y
= 5xy * (x – 3y)
(vii) 10a² – 15b² + 20c²
Factorise:
5 * 2 * a * a – 5 * 3 * b * b + 5 * 4 * c * c
= 5 * (2a² – 3b² + 4c²)
(viii) –4a² + 4ab – 4ca
Factorise:
-4 * a * a + 4 * a * b – 4 * c * a
= 4a * (-a + b – c)
(ix) x²yz + xy²z + xyz²
Factorise:
x * x * y * z + x * y * y * z + x * y * z * z
= xyz * (x + y + z)
(x) ax²y + bxy² + cxyz
Factorise:
x * a * x * y + x * b * y * y + x * c * y * z
= xy * (ax + by + cz)
3. Factorise.
(i) x² + x y + 8x + 8y (ii) 15 xy – 6x + 5y – 2 (iii) ax + bx – ay – by (iv) 15 pq + 15 + 9q + 25p (v) z – 7 + 7 x y – x y z
(i) x² + x y + 8x + 8y
Factorise:
= x * (x + y) + 8 * (x + y)
= (x + y)(x + 8)
(ii) 15 xy – 6x + 5y – 2
Factorise:
= 3x * (5y – 2) + 1 * (5y – 2)
= (5y – 2)(3x + 1)
(iii) ax + bx – ay – by
Factorise:
= a * (x – y) + b * (x – y)
= (x – y)(a + b)
(iv) 15 pq + 15 + 9q + 25p
Factorise:
= 3 * (5pq + 5 + 3q + 25/3 * p)
= 3(5 * (pq + 1) + 3q + 25/3 * p)
(v) z – 7 + 7xy – xyz
Factorise:
= 1 * (z – 7) + xy * (7 – z)
= (z – 7)(1 – xy)
