Find the GCD of the following $x^{2} + 3 x y + 2 y^{2}, x^{2} + 5 x y + 6 y^{2}$

x + 2 y

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x - 2 y

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2 x + 2 y

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None of these

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Solution

The correct option is D $x + 2 y$
We first factorize the given polynomials $x^{2} + 3 x y + 2 y^{2}$ and $x^{2} + 5 x y + 6 y^{2}$ as shown below:

$x^{2} + 3 x y + 2 y^{2} = x^{2} + x y + 2 x y + 2 y^{2} = x (x + y) + 2 y (x + y) = (x + y) (x + 2 y)$

$x^{2} + 5 x y + 6 y^{2} = x^{2} + 2 x y + 3 x y + 6 y^{2} = x (x + 2 y) + 3 y (x + 2 y) = (x + 3 y) (x + 2 y)$

The common factor of $x^{2} + 3 x y + 2 y^{2}$ and $x^{2} + 5 x y + 6 y^{2}$ is $(x + 2 y)$ , therefore, the GCD is $x + 2 y$ .

Hence, the greatest common divisor of $x^{2} + 3 x y + 2 y^{2}$ and $x^{2} + 5 x y + 6 y^{2}$ is $x + 2 y$ .

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