Factorise the expression:
$x^{4} + y^{4} - 3 x^{2} y^{2} .$

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

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

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

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

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Solution

The correct option is D $(x^{2} - y^{2} + x y) (x^{2} - y^{2} - x y)$
Given, the expression is
$x^{4} + y^{4} - 3 x^{2} y^{2} .$
the above expression can be written as,
$(x^{2})^{2} + (y^{2})^{2} - 2 x^{2} y^{2} - x^{2} y^{2}$
$= (x^{2})^{2} - 2 x^{2} y^{2} + (y^{2})^{2} - x^{2} y^{2}$
Using the identity
$(a - b)^{2} = a^{2} - 2 a b + b^{2}, we get,$
$(x^{2} - y^{2})^{2} - (x y)^{2}$
$[Using identity: a^{2} - b^{2} = (a - b) (a + b)]$
$= (x^{2} - y^{2} + x y) (x^{2} - y^{2} - x y)$

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x^{4} + y^{4} - 3 x^{2} y^{2}

Question 58 (g)

Subtract:

$x^{3} y^{2} + 3 x^{2} y^{2} - 7 x y^{3}$ from $x^{4} + y^{4} + 3 x^{2} y^{2} - x y^{3} .$

Question 56 (b)

Simplify the following by combining the like terms and then write whether the expression is a monomial, a binomial or a trinomial.

$x^{4} + 3 x^{3} y + 3 x^{2} y^{2} - 3 x^{3} y - 3 x y^{2} + y^{4} - 3 x^{2} y^{2}$

16 x^{4} - y^{4}

x^{4} - y^{4}

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