Factorise the following: $(3 x + 5 y)^{3} - (2 x - y)^{3}$

(6 y) (19 x^{2} + 3 x + 4 y^{2})

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(x + 6 y) (19 x^{2} + 33 x y + 21 y^{2})

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(x y + 2) (19 x^{2} + x y + y^{2})

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(x + 6 y) (x^{2} + 13 x y + 12 y^{2})

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Solution

The correct option is B $(x + 6 y) (19 x^{2} + 33 x y + 21 y^{2})$
$(3 x + 5 y)^{3} - (2 x - y)^{3}$
Using, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$
$= [(3 x + 5 y) - (2 x - y)] [(3 x + 5 y)^{2} + (3 x + 5 y) (2 x - y) + (2 x - y)^{2}]$
$= [3 x + 5 y - 2 x + y] [9 x^{2} + 30 x y + 25 y^{2} + 6 x^{2} - 3 x y + 10 x y - 5 y^{2} + 4 x^{2} - 4 x y + y^{2})]$
$= [x + 6 y] [19 x^{2} + 33 x y + 21 y^{2}]$

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(i) $x^{2} + y^{2} + z^{2} - x y + x + y z by x + y - z$

(ii) $x^{2} + 4 y^{2} + z^{2} + 2 x y + x z - 2 y z by x - 2 y - z$

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