Factorise each of the following:
$1) {8 a}^{3} + b^{3} + 12 a^{2} b + 6 a b^{2}$
$2) 64 m^{3} - 343 n^{3}$ [4 MARKS]

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Solution

Factorisation: 2 Marks each

$1) {8 a}^{3} + b^{3} + 12 a^{2} b + 6 a b^{2}$

$= {(2 a)}^{3} + b^{3} + 6 a b (2 a + b)$

$= {(2 a)}^{3} + b^{3} + 3 (2 a) (b) (2 a + b)$ ...eq(i)

$x^{3} + y^{3} + 3 x y (x + y) = {(x + y)}^{3}$

$x = 2 a, y = b$ [comparing eq(i) with the above identity]

${(2 a)}^{3} + b^{3} + 3 (2 a) (b) (2 a + b) = {(2 a + b)}^{3}$

$= (2 a + b) (2 a + b) (2 a + b)$

$2) 64 m^{3} - 343 n^{3}$

$= {(4 m)}^{3} - {(7 n)}^{3}$ ...eq(i)

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$

$a = 4 m, b = 7 n$ [comparing eq(i) with the above identity]

${(4 m)}^{3} - {(7 n)}^{3} = (4 m - 7 n) ({(4 m)}^{2} + (4 m) (7 n) + {(7 n)}^{2})$

$= (4 m - 7 n) (16 m^{2} + 28 m n + 49 n^{2})$

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