Factor polynomial $$ 32w^{2}-16w+2 $$

$$ 32w^{2}-16w+2 = 2( 4w-1 )^{ 2 } $$

Step 1 :

After factoring out $ 2 $ we have:

$$ 32w^{2}-16w+2 = 2 ( 16w^{2}-8w+1 ) $$

Step 2 :

Both the first and third terms are perfect squares.

$$ 16x^2 = \left( \color{blue}{ 4w } \right)^2 ~~ \text{and} ~~ 1 = \left( \color{red}{ 1 } \right)^2 $$

The middle term ( $ -8x $ ) is two times the product of the terms that are squared.

$$ -8x = - 2 \cdot \color{blue}{4w} \cdot \color{red}{1} $$

We can conclude that the polynomial $ 16w^{2}-8w+1 $ 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 = 4w } $ and $ \color{red}{ B = 1 } $ so,

$$ 16w^{2}-8w+1 = ( \color{blue}{ 4w } - \color{red}{ 1 } )^2 $$

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