Divide $3 x^{4} - 5 x^{3} y + 6 x^{2} y^{2} - 3 x y^{3} + y^{4}$ by $x^{2} - x y + y^{2}$ .
What should be subtracted from the quotient to make it a perfect square?

2 x^{2}

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x^{2}

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y^{2}

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-xy

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Solution

The correct option is A $2 x^{2}$

We divide $x^{2} - x y + y^{2}$ by $3 x^{4} - 5 x^{3} y + 6 x^{2} y^{2} - 3 x y^{3} + y^{4}$ as shown in the above division.

Therefore, the quotient is $3 x^{2} - 2 x y + y^{2}$ .

Now, to make the quotient a perfect square, we must subtract $2 x^{2}$ as follows:

$3 x^{2} - 2 x y + y^{2} - 2 x^{2} = (3 x^{2} - 2 x^{2}) - 2 x y + y^{2} = x^{2} - 2 x y + y^{2} = x^{2} + y^{2} - 2 x y = (x - y)^{2} (∵ (a - b)^{2} = a^{2} + b^{2} - 2 a b)$

Hence, $2 x^{2}$ should be subtracted from the quotient to make it a perfect square.

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