Factor polynomial $$ x^{6}+3x^{4}-x^{2}-3 $$

$$ x^{6}+3x^{4}-x^{2}-3 = ( x-1 )( x+1 )( x^{2}+1 )( x^{2}+3 ) $$

Step 1 :

To factor $ x^{6}+3x^{4}-x^{2}-3 $ we can use factoring by grouping:

Group $ \color{blue}{ x^{6} }$ with $ \color{blue}{ 3x^{4} }$ and $ \color{red}{ -x^{2} }$ with $ \color{red}{ -3 }$ then factor each group.

$$ \begin{aligned} x^{6}+3x^{4}-x^{2}-3 = ( \color{blue}{ x^{6}+3x^{4} } ) + ( \color{red}{ -x^{2}-3 }) &= \\ &= \color{blue}{ x^{4}( x^{2}+3 )} + \color{red}{ -1( x^{2}+3 ) } = \\ &= (x^{4}-1)(x^{2}+3) \end{aligned} $$

Step 2 :

Rewrite $ x^{4}-1 $ as:

$$ x^{4}-1 = (x^{2})^2 - (1)^2 $$

Now we can apply the difference of squares formula.

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

After putting $ I = x^{2} $ and $ II = 1 $ , we have:

$$ x^{4}-1 = (x^{2})^2 - (1)^2 = ( x^{2}-1 ) ( x^{2}+1 ) $$

Step 3 :

Rewrite $ x^{2}-1 $ as:

$$ x^{2}-1 = (x)^2 - (1)^2 $$

Now we can apply the difference of squares formula.

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

After putting $ I = x $ and $ II = 1 $ , we have:

$$ x^{2}-1 = (x)^2 - (1)^2 = ( x-1 ) ( x+1 ) $$

Polynomial Factoring Calculator