A factor of $a^{2} - 2 a b + b^{2} - c^{2}$ is ___________.

$a - b$

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$a - b - c$

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$a + b + c$

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$a + b - c$

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Solution

The correct option is B
$a - b - c$

$a^{2} - 2 a b + b^{2} - c^{2} = a^{2} - a b - a b + b^{2} - c^{2} = a (a - b) - b (a - b) - c^{2}$
$= (a - b) (a - b)$ $- c^{2}$
$= (a - b)^{2} - c^{2}$

Now, using the identity $x^{2} - y^{2} = (x + y) (x - y)$ we get

$(a - b)^{2} - c^{2}$ = $[(a - b) + c] [(a - b) - c]$

Hence, the factors of $(a^{2} - 2 a b + b^{2} - c^{2})$ are $(a - b + c)$ and $(a - b - c)$

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a^{2} - 2 a b - c^{2} + b^{2}

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a^{2} + b^{2} - c^{2} - d^{2} + 2 a b - 2 c d

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