Factorize $4 x^{2} + y^{2} + z^{2} - 4 x y - 2 y z + 4 x z$

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Solution

Factorizing the given expression:
The given expression is $4 x^{2} + y^{2} + z^{2} - 4 x y - 2 y z + 4 x z$ .
Use identity $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$ to expand the expression.
So, expression can be rewritten as:
$4 x^{2} + y^{2} + z^{2} - 4 x y - 2 y z + 4 x z = {(2 x)}^{2} + {(- y)}^{2} + {(z)}^{2} + 2 (2 x) (- y) + 2 (- y) (z) + 2 (2 x) (z)$
Here, $a = 2 x, b = - y, c = z$ .
So, $4 x^{2} + y^{2} + z^{2} - 4 x y - 2 y z + 4 x z$ $= {[2 x + (- y) + (z)]}^{2} [Using identity]$
Thus, the expression, $4 x^{2} + y^{2} + z^{2} - 4 x y - 2 y z + 4 x z = {(2 x - y + z)}^{2}$ .

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