Find the product of $(5 x - y + z) (5 x - y + z)$

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Solution

We know $(5 x - y + z) (5 x - y + z)$ $= (5 x - y + z)^{2}$
Using identity $(x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2 x y + 2 y z + 2 x z$
Here $x = 5 x, y = - y$ and $z = z$
Therefore, $(5 x - y + z)^{2}$ $
$= (5 x)^{2} + (- y)^{2} + (z)^{2} + 2 (5 x) (- y) + 2 (- y) (z) + 2 (5 x) (z)$
$= 25 x^{2} + y^{2} + z^{2} - 10 x y - 2 y z + 10 x z$

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