Factor polynomial $$ x^{3}-7x^{2}-16x+112 $$

$$ x^{3}-7x^{2}-16x+112 = ( x-4 )( x+4 )( x-7 ) $$

Step 1 :

To factor $ x^{3}-7x^{2}-16x+112 $ we can use factoring by grouping:

Group $ \color{blue}{ x^{3} }$ with $ \color{blue}{ -7x^{2} }$ and $ \color{red}{ -16x }$ with $ \color{red}{ 112 }$ then factor each group.

$$ \begin{aligned} x^{3}-7x^{2}-16x+112 = ( \color{blue}{ x^{3}-7x^{2} } ) + ( \color{red}{ -16x+112 }) &= \\ &= \color{blue}{ x^{2}( x-7 )} + \color{red}{ -16( x-7 ) } = \\ &= (x^{2}-16)(x-7) \end{aligned} $$

Step 2 :

Rewrite $ x^{2}-16 $ as:

$$ x^{2}-16 = (x)^2 - (4)^2 $$

Now we can apply the difference of squares formula.

$$ I^2 - II^2 = (I - II)(I + II) $$

After putting $ I = x $ and $ II = 4 $ , we have:

$$ x^{2}-16 = (x)^2 - (4)^2 = ( x-4 ) ( x+4 ) $$

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