Factor polynomial $$ 2x^{4}-24x^{2}+72 $$

$$ 2x^{4}-24x^{2}+72 = 2( x^{2}-6 )^{ 2 } $$

Step 1 :

After factoring out $ 2 $ we have:

$$ 2x^{4}-24x^{2}+72 = 2 ( x^{4}-12x^{2}+36 ) $$

Step 2 :

Both the first and third terms are perfect squares.

$$ x^4 = \left( \color{blue}{ x^{2} } \right)^2 ~~ \text{and} ~~ 36 = \left( \color{red}{ 6 } \right)^2 $$

The middle term ( $ -12x^2 $ ) is two times the product of the terms that are squared.

$$ -12x^2 = - 2 \cdot \color{blue}{x^{2}} \cdot \color{red}{6} $$

We can conclude that the polynomial $ x^{4}-12x^{2}+36 $ is a perfect square trinomial, so we will use the formula below.

$$ A^2 - 2AB + B^2 = (A - B)^2 $$

In this example we have $ \color{blue}{ A = x^{2} } $ and $ \color{red}{ B = 6 } $ so,

$$ x^{4}-12x^{2}+36 = ( \color{blue}{ x^{2} } - \color{red}{ 6 } )^2 $$

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