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(a+b)(a-b) Expansion and Visualisation

Find the HCF ...

Question

Find the HCF of the following pair of polynomials:
$x^{2} - x y - 2 y^{2}$ and $x^{2} + 3 x y + 2 y^{2}$

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Solution

We know that $H C F$ is the highest common factor.

Factorise $x^{2} - x y - 2 y^{2}$ as follows:

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

Now, factorise $x^{2} + 3 x y + 2 y^{2}$ as follows:

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

Since the common factor between the polynomials $x^{2} - x y - 2 y^{2}$ and $x^{2} + 3 x y + 2 y^{2}$ is $(x + y)$ .

Hence, the $H C F$ is $(x + y)$ .

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