Factorize the given polynomial expression

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

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Solution

Given that,
$x^{4} + x^{2} y^{2} + y^{4}$

Let, $x^{2} = a, y^{2} = b$
So the the expression becomes,
$= a^{2} + a b + b^{2}$
$= a^{2} + a b + b^{2} + a b - a b$
$= a^{2} + 2 a b + b^{2} - a b$
$= (a + b)^{2} - a b$ $[∵ (a + b)^{2} = a^{2} + 2 a b + b^{2}]$

By putting the values of a & b we get,
$= (x^{2} + y^{2})^{2} - x^{2} y^{2}$
$= (x^{2} + y^{2})^{2} - (x y)^{2}$

this is in the form,
$p^{2} - q^{2} = (p + q) (p - q)$

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

$∴ x^{4} + x^{2} y^{2} + x^{4} = (x^{2} + x y + y^{2}) (x^{2} - x y + y^{2})$

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i) x⁴ - y⁴+ x²- y²

(x^{4} - y^{4}) (x^{4} + x^{2} y^{2} + y^{4}), x^{2} - y^{2}, x^{4} - y^{4}

The simplified form of the expression given below is :-

$\begin{matrix} y^{4} - x^{4} - ------ - - y^{3} - - - \frac{x (x + y) x}{y^{2} - x y + x^{2}} \end{matrix}$

\frac{\frac{y^{4} - x^{4}}{x (x + y)} - \frac{y^{3}}{x}}{y^{2} - x y + x^{2}}

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

(x^{4} - y^{4}) : (x + y)^{4}

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