Factorise the following expressions:
(i) a3 - 27
(ii) 1 - 27x3
(iii) 8x3 - (2x - 3y)3
(iv) a8 - a2b6
(v) a3 - 5
b3
(i) a3 - 27
= a3 - (3)3
= (a - 3) (a2 + 3a + 9)
(ii) 1 - 27x3 = 13 - (3x)3
= (1 - 3x) [12 + 1 × 3x + (3x)2]
= (1 - 3x) (1 + 3x + 9x2)
(iii) 8x3 - (2x - 3y)3 = (2x)3 - (2x - 3y)3
= [2x - (2x - 3y)][(2x)2 + 2x(2x - 3y) + (2x - 3y)2]
= (2x-2x+ 3y) (4x2 + 4x2- 6xy + 4x2+ 9y2-12xy)
= 3y(12x2 + 9y2 - 18xy)
= 3y൳(4x2 + 3y2 - 6xy)
= 9y(4x2 + 3y2 - 6xy)
(iv) a8 - a2b6 = a2(a6 - b6)
= a2[(a3)2 - (b3)2]
= a2 (a3 + b3) (a3 - b3)
= a2 (a + b) (a2 - ab + b2) (a - b) (a2 + ab + b2)
= a2 (a + b) (a - b) (a2 - ab + b2) (a2 + ab + b2)
(v) a3 - 5
b3 = a3 - (b)3
= (a-b)(a2+ ab+5b2)
= a3 - (3)3
= (a - 3) (a2 + 3a + 9)
(ii) 1 - 27x3 = 13 - (3x)3
= (1 - 3x) [12 + 1 × 3x + (3x)2]
= (1 - 3x) (1 + 3x + 9x2)
(iii) 8x3 - (2x - 3y)3 = (2x)3 - (2x - 3y)3
= [2x - (2x - 3y)][(2x)2 + 2x(2x - 3y) + (2x - 3y)2]
= (2x-2x+ 3y) (4x2 + 4x2- 6xy + 4x2+ 9y2-12xy)
= 3y(12x2 + 9y2 - 18xy)
= 3y൳(4x2 + 3y2 - 6xy)
= 9y(4x2 + 3y2 - 6xy)
(iv) a8 - a2b6 = a2(a6 - b6)
= a2[(a3)2 - (b3)2]
= a2 (a3 + b3) (a3 - b3)
= a2 (a + b) (a2 - ab + b2) (a - b) (a2 + ab + b2)
= a2 (a + b) (a - b) (a2 - ab + b2) (a2 + ab + b2)
(v) a3 - 5
b3 = a3 - (b)3= (a-
b)(a2+ ab+5b2) AI is thinking…
Couldn't generate an explanation.
Generated by AI. May contain inaccuracies — always verify with your textbook.
+ 8b3
a3 + 3
b3
ab4