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Factorization by Grouping Method

Factorize:-a3...

Question

Factorize:-
a³ + 3b³ -64c³+24abc
(36x²-1) + (1+6x)²
27y³+125y³
25x⁴-4y⁴
64a²+144a+81

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Solution

$(1) . a^{3} + 8 b^{3} - 64 c^{3} + 24 a b c = {(a)}^{3} + {(2 b)}^{3} + {(- 4 c)}^{3} - 3 (a) (2 b) (- 4 c) = [a + 2 b + (- 4 c)] [{(a)}^{2} + {(2 b)}^{2} + {(- 4 c)}^{2} - (a) (2 b) - (2 b) (- 4 c) - (- 4 c) (a)] = (a + 2 b - 4 c) (a^{2} + 4 b^{2} + 16 c^{2} - 2 a b + 8 b c + 4 c a) (2) . (36 x^{2} - 1) + {(6 x + 1)}^{2} = [{(6 x)}^{2} - {(1)}^{2}] + (6 x + 1) (6 x + 1) = (6 x - 1) (6 x + 1) + (6 x + 1) (6 x + 1) = (6 x + 1) [(6 x - 1) + (6 x + 1)] = 12 x (6 x + 1) (3) . 27 y^{3} + 125 z^{3} = {(3 y)}^{3} + {(5 z)}^{3} = (3 y + 5 z) [{(3 y)}^{2} + {(5 z)}^{2} - (3 y) (5 z)] = (3 y + 5 z) (9 y^{2} + 25 z^{2} - 15 y z) (4) . 25 x^{4} - 4 y^{4} = {(5 x^{2})}^{2} - {(2 y^{2})}^{2} = (5 x^{2} + 2 y^{2}) (5 x^{2} - 2 y^{2}) (5) . 64 a^{2} + 144 a + 81 = {(8 a)}^{2} + 2 (8 a) (9) + {(9)}^{2} = {(8 a + 9)}^{2} = (8 a + 9) (8 a + 9)$

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