**1. Find the common factors of the given terms.**

**(i) 12x, 36**

**Solution:**

Factors of 12x and 36

12x = 2×2×3×x

36 = 2×2×3×3

Common factors are 2, 2, 3

Hence, common factors are 2×2×3 = 12

**(ii) 2y, 22xy**

**Solution:**

Factors of 2y and 22xy

2y = 2 × y

22xy = 2×11× x ×y

Common factors = 2, y

Hence common factors are 2 × y = 2y

**(iii) 14 pq, 28p ^{2}q^{2}**

**Solution:**

Factors of 14pq and 28p^{2}q^{2}

14pq = 2 x 7 x p x q

28p^{2}q^{2} = 2 x 2 x7 x p x p x q x q

Common factors are 2, 7 , p , q

Hence, common factors are 2 x 7 x p x q = 14pq

**(iv) 2x, 3x ^{2}, 4**

**Solution:**

Factors of 2x, 3x^{2}and 4

2 x = 2 × x

3 x 2= 3 × x × x

4 = 2 × 2

Hence, common factor is 1.

**(v) 6 abc, 24ab ^{2}, 12a^{2}b**

**Solution:**

Factors of 6abc, 24ab^{2} and 12a^{2}b

6abc = 2 × 3 × a × b × c

24ab^{2} = 2 × 2 × 2 × 3 × a × b × b

12 a^{2} b = 2 × 2 × 3 × a × a × b

Common factors are 2, 3, a, b

Hence, common factors are 2 × 3 × a × b = 6ab

**(vi) 16 x ^{3}, – 4x^{2}, 32 x**

**Solution:**

Factors of 16x^{3} , -4x^{2}and 32x

16 x^{3} = 2 × 2 × 2 × 2 × x × x × x

– 4x^{2} = -1 × 2 × 2 × x × x

32x = 2 × 2 × 2 × 2 × 2 × x

Common factors are 2,2, x

Hence, common factors are 2×2×x = 4x

**(vii) 10 pq, 20qr, 30 rp**

**Solution:**

Factors of 10 pq, 20qr and 30rp

10 p q = 2 × 5 × p × q

20 q r = 2 × 2 × 5 × q × r

30 r p= 2× 3 × 5 × r × p

Common factors are 2, 5

Hence, common factors are 2×5 = 10

**(viii) 3x ^{2}y^{3}, 10x^{3}y^{2}, 6x^{2}y^{2}z**

**Solution:**

Factors of 3x^{2}y^{3}, 10x^{3}y^{2} and 6x^{2}y^{2}z

3 x^{2} y^{3} = 3 × x × x × y × y × y

10x^{3 }y^{2} = 2 × 5 × x × x × x × y × y

6x^{2}y^{2}z = 3 × 2 × x × x × y × y × z

Common factors are x^{2}, y^{2}

Hence, common factors are x^{2} × y^{2} = x^{2}y^{2}

**2.Factorise the following expressions**

**(i) 7x–42**

**Solution:**

Take 7 as common

= 7 (x – 6)

**(ii) 6p–12q**

**Solution:**

Take 6 as common

= 6(p – 2q)

**(iii) 7a ^{2}+ 14a**

**Solution:**

Take 7a as common

=7a (a + 2)

**(iv) -16z+20 z ^{3}**

**Solution:**

Take 4z as common

= 4z ( -4 + 5 z^{2})

**(v) 20l ^{2}m+30alm**

**Solution:**

Take 10 l m as common

= 10 l m (2 l + 3a)

**(vi) 5x ^{2}y-15xy^{2}**

**Solution:**

Take 5xy as common

= 5xy (x – 3y)

(**vii) 10a ^{2}-15b^{2}+20c^{2}**

**Solution:**

Take 5 as common

= 5 (2 a^{2} – 3 b^{2} + 4 c^{2})

**(viii) -4a ^{2}+4ab–4 ca**

**Solution:**

Take 4a as common

= 4a (-a + b – c)

**(ix) x ^{2}yz+xy^{2}z +xyz^{2}**

**Solution:**

Take xyz as common

= xyz ( x + y + z)

**(x) ax ^{2}y+bxy^{2}+cxyz**

**Solution:**

Take xy as common

= xy (ax + by + z)

**3. Factorise.**

**(i) x ^{2}+xy+8x+8y**

**Solution:**

= (x^{2}+xy) + (8x+8y)

= x (x + y) + 8 (x +y)

= (x + y) (x + 8)

Hence, required factors: (x + y) (x + 8)

**(ii) 15xy–6x+5y–2**

**Solution:**

= (15xy–6x) + (5y–2)

= 3x (5y – 2) + (5y – 2)

= (5y – 2) (3x +1)

Hence, required factors: (5y – 2) (3x +1)

**(iii) ax+bx–ay–by**

**Solution:**

= (ax – ay) – (bx –by)

= a (x – y) – b (x – y)

= (a – b) (x – y)

Hence required factors: (a – b) (x – y)

**(iv) 15pq + 15 + 9q + 25p**

**Solution**:

= (15pq + 25p) + (9 q + 15)

= 5p (3q + 5) + 3(3q +5)

= (5p + 3) (3q + 5)

Hence, required factors: (5p + 3) (3q + 5)

**(v) z–7+7xy–xyz**

**Solution:**

= (-xyz + 7xy) + (z – 7)

= -xy (z – 7) + (z – 7)

= (-xy + 1) (z – 7)

Hence, required factors: (-xy + 1) (z – 7)

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