Factorise :

$(i) 2 a^{3} - 50 a (i i) 54 a^{2} b^{2} - 6 (i i i) 64 a^{2} b - 144 b^{3} (i v) (2 x - y)^{3} - (2 x - y) (v) x^{2} - 2 x y + y^{2} - z^{2} (v i) x^{2} - y^{2} - 2 y z - z^{2} (v i i) 7 a^{5} - 567 a (v i i i) 5 x^{2} - \frac{20 x^{4}}{9}$

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Solution

$(i) 2 a^{3} - 50 a = 2 a (a^{2} - 25) = 2 a (a^{2} - 5^{2}) = 2 a (a - 5) (a + 5) (i i) 54 a^{2} b^{2} - 6 = 6 (9 a^{2} b^{2} - 1) = 6 [(3 a b)^{2} - (1)^{2}] = 6 (3 a b - 1) (3 a b + 1) (i i i) 64 a^{2} b - 144 b^{3} = 16 b (4 a^{2} - 9 b^{2}) = 16 b [(2 a)^{2} - (3 b)^{2}] = 16 b (2 a + 3 b) (2 a - 3 b) (i v) (2 x - y)^{3} - (2 x - y) = (2 x - y) [(2 x - y)^{2} - 1] = (2 x - y) (2 x - y - 1) (2 x - y + 1) (v) x^{2} - 2 x y + y^{2} - z^{2} = (x^{2} - 2 x y + y^{2}) - z^{2} = (x - y)^{2} - (z)^{2} = (x - y - z) (x - y + z) (v i) x^{2} - y^{2} - 2 y z - z^{2} = x^{2} - (y^{2} + 2 y z + z^{2}) = x^{2} - (y + z)^{2} = (x - y - z) (x + y + z) (v i i) 7 a^{5} - 567 a = 7 a (a^{4} - 81) = 7 a (a^{2})^{2} - (9)^{2} = 7 a (a^{2} + 9) (a^{2} - 9) = 7 a (a^{2} + 9) ((a)^{2} - (3)^{2}) = 7 a (a^{2} + 9) (a + 3) (a - 3) (v i i i) 5 x^{2} - \frac{20 x^{4}}{9} = 5 x^{2} [1 - \frac{4 x^{2}}{9}] = 5 x^{2} [1 - \frac{2 x}{3}] [1 + \frac{2 x}{3}]$

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