Question 8
Factorise each of the following:
$(i) 8 a^{3} + b^{3} + 12 a^{2} b + 6 a b^{2} (i i) 8 a^{3} - b^{3} - 12 a^{2} b + 6 a b^{2} (i i i) 27 - 125 a^{3} - 135 a + 225 a^{2} (v i) 64 a^{3} - 27 b^{3} - 144 a^{2} b + 108 a b^{2} (v) 27 p^{3} - \frac{1}{216} - \frac{9}{2} p^{2} + \frac{1}{4} p$

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Solution

$(i) 8 a^{3} + b^{3} + 12 a^{2} b + 6 a b^{2} = (2 a)^{3} + b^{3} + 3 (2 a) (b) (2 a + b) U s i n g i d e n t i t y (a + b)^{3} = a^{3} + b^{3} + 3 a b (a + b) = (2 a + b)^{3} = (2 a + b) (2 a + b) (2 a + b)$

$(i i) 8 a^{3} - b^{3} - 12 a^{2} b + 6 a b^{2} = (2 a)^{3} + (- b)^{3} + 3 (2 a) (- b) (2 a - b) U s i n g i d e n t i t y (a - b)^{3} = a^{3} - b^{3} - 3 a b (a - b) = (2 a - b)^{3} = (2 a - b)^{3} = (2 a - b) (2 a - b) (2 a - b)$

$(i i i) 27 - 125 a^{3} - 135 a + 225 a^{2} = 3^{3} + (- 5 a)^{3} + 3 \times (3) (- 5 a) (3 - 5 a) U s i n g i d e n t i t y (a - b)^{3} = a^{3} - b^{3} - 3 a b (a - b) = (3 - 5 a)^{3} = (3 - 5 a) (3 - 5 a) (3 - 5 a)$

$(i v) 64 a^{3} - 27 b^{3} - 144 a^{2} b + 108 a b^{2} = (4 a)^{3} + (- 3 b)^{3} + 3 (4 a) \times (- 3 b) (4 a - 3 b) U s i n g i d e n t i t y (a - b)^{3} = a^{3} - b^{3} - 3 a b (a - b) = (4 a - 3 b)^{3} = (4 a - 3 b) (4 a - 3 b) (4 a - 3 b)$

$(v) 27 p^{3} - \frac{1}{216} - \frac{9}{2} p^{2} + \frac{1}{4} p = (3 p)^{3} + {(- \frac{1}{6})}^{3} + 3 (3 p) (- \frac{1}{6}) (3 p - \frac{1}{6}) U s i n g i d e n t i t y (a - b)^{3} = a^{3} - b^{3} - 3 a b (a - b) = {(3 p - \frac{1}{6})}^{3} = (3 p - \frac{1}{6}) (3 p - \frac{1}{6}) (3 p - \frac{1}{6})$

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Similar questions

Factorise each of the following:

(i) (ii)

(iii) (iv)

(v)

8 a^{3} + b^{3} + 12 a^{b} + 6 a b^{2}

8 a^{3} - b^{3} - 12 a^{2} b + 6 a b^{2}

27 - 125 a^{3} - 135 a + 225 a^{2}

64 a^{3} - 27 b^{3} - 144 a^{2} b + 108 a b^{2}

27 p^{3} - \frac{1}{216} - \frac{9}{2} p^{2} + \frac{1}{4} p

8 a^{3} + b^{3} + 12 a^{2} b + 6 a b^{2}

Factorise: $8 a^{3} - b^{3} - 12 a^{2} b + 6 a b^{2}$

Factorise : 8a³ + b³ + 12a²b + 6ab²

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