What is the HCF of $4 x^{3} + 3 x^{2} y - 9 x y^{2} + 2 y^{3}$ and $x^{2} + x y - 2 y^{2}$ ?

x - 2 y

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x - y

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(x + 2 y) (x - y)

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(x - 2 y) (x - y)

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Solution

The correct option is C $(x + 2 y) (x - y)$
Let $f (x, y) = 4 x^{2} + 3 x^{2} y - 9 x y^{2} + 2 y^{3}$
and $g (x, y) = x^{2} + x y - 2 y^{2}$
Clearly $(x - y) = 0$ or $x = y$ satisfy both equation, since for $x = y$ makes both equations zero.
Hence, $(x - y)$ is the H.C.F.
Similarly $(x + 2 y) = 0$ i.e., $x = - 2 y$ also makes both equations zero.
Thus, $(x + 2 y) (x - y)$ is the H.C.F.

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