Exercise 12.1

1. Find the common factors of the given terms in each.

(i) 8x, 24

Solution:

8x = × × ×

24 = × × ×

Common factors: × × =

(ii) 3a, 21ab

Solution:

3a = ×

21ab = × × ×

Common factors: × =

(iii) 7xy, 35x2y3

Solution:

7xy = × ×

35x2y3 = × × × × × ×

Common factors: × × =

(iv) 4m2, 6m2, 8m3

Solution:

4m2 = × × mm2 × mm2

6m2 = × × ×

8m3 = × × × × ×

Common factors: × × = 2m22m3

(v) 15p, 20qr, 25rp

Solution:

15p = × ×

20qr = × × × ×

25rp = × × ×

Common factor:

(vi) 4x2, 6xy, 8y2x

Solution:

4x2 = × × ×

6xy = × × ×

8y2x = × × × × ×

Common factors: × =

(vii) 12x2y, 18xy2

Solution:

12x2y = × × × × ×

18xy2 = × × × [x] × ×

Common factors: × × × =

2. Factorise the following expressions

(i) 5x2 – 25xy

Solution:

5x2 - 25xy = (x - 5y)

(ii) 9a2 – 6ax

Solution:

9a2 - 6ax = 3a(3a - 2x) = (3a - 2x)

(iii) 7p2 + 49pq

Solution:

7p2 + 49pq = (p + 7q)

(iv) 36a2b – 60a2bc

Solution:

36a2b – 60a2bc = 12a2b12ab(3 - 5c)

(v) 3a2bc + 6ab2c + 9abc2

Solution:

3a2bc + 6ab2c + 9abc2 = (a + 2b + 3c)

(vi) 4p2 + 5pq – 6pq2

Solution:

4p2 + 5pq – 6pq2 = (4p + 5q - 6q²)

(vii) ut + at2

Solution:

ut + at2 = (u + at)

3. Factorise the following :

(i) 3ax – 6xy + 8by – 4ab

Solution:

3ax - 6xy + 8by - 4ab = (a - 2y) - (a - 2y)

= (a - 2y)( - )

(ii) x3 + 2x2 + 5x + 10

Solution:

x3 + 2x2 + 5x + 10 = x2x4(x + 2) + (x + 2)

= (x + 2)(x22x + 52)

(iii) m2 – mn + 4m – 4n

Solution:

m2 – mn + 4m – 4n = (m - n) + (m - n)

= (m - n)( + )

(iv) a3 – a2b2 – ab + b3

Solution:

a3 – a2b2 – ab + b3 = a2a3(a - b2) - (a - b2)

= (a - b2)(a2a - bb2)

(v) p2q – pr2 – pq + r2

Solution:

p2q – pr2 – pq + r2 = (pq - r2) - (pq - r2)

= (pq - r2)( - )