Resolve $a^{4} - 18 a^{2} b^{2} + b^{4}$ into factors.

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Solution

Given equation is $a^{4} - 18 a^{2} b^{2} + b^{4}$

$a^{4} - 18 a^{2} b^{2} + b^{4}$ $= a^{4} - 2 \times 9 a^{2} b^{2} + 81 b^{4} - 80 b^{4}$
$= {(a^{2} - 9 b^{2})}^{2} - {(4 \sqrt{5} b^{2})}^{2}$

As we know that, $a^{2} - b^{2} = (a + b) (a - b)$

$a^{4} - 18 a^{2} b^{2} + b^{4}$ $= (a^{2} - 9 b^{2} + 4 \sqrt{5} b^{2}) (a^{2} - 9 b^{2} - 4 \sqrt{5} b^{2})$

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