If $4 x - y$ is a multiple of $3$ , show that $4 x^{2} + 7 x y - 2 y^{3}$ is divisible by $9$ .

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Solution

Given :
$\Rightarrow 4 x^{2} + 7 x y - 2 y^{2}$

$= 4 x^{2} + 8 x y - x y - 2 y^{2}$

$= 4 x (x + 2 y) - y (x + 2)$

$= (4 x - y) (x + 2 y)$

given that $4 x - y$ is divisible by $3$

$\Rightarrow 4 x - y + 3 y - 3 x = x + 2 y$

$x + 2 y$ is divisible by $3$ as we have only added and subtracted integer multiples of $3$

$\Rightarrow (4 x - y) (x + 2 y)$ is divisible by $3 \times 3 = 9$

Hence proved.

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