Factor polynomial $$ 9x^{6}-18x^{4}-x^{2}+2 $$

$$ 9x^{6}-18x^{4}-x^{2}+2 = ( 3x^{2}-1 )( 3x^{2}+1 )( x^{2}-2 ) $$

Step 1 :

To factor $ 9x^{6}-18x^{4}-x^{2}+2 $ we can use factoring by grouping:

Group $ \color{blue}{ 9x^{6} }$ with $ \color{blue}{ -18x^{4} }$ and $ \color{red}{ -x^{2} }$ with $ \color{red}{ 2 }$ then factor each group.

$$ \begin{aligned} 9x^{6}-18x^{4}-x^{2}+2 = ( \color{blue}{ 9x^{6}-18x^{4} } ) + ( \color{red}{ -x^{2}+2 }) &= \\ &= \color{blue}{ 9x^{4}( x^{2}-2 )} + \color{red}{ -1( x^{2}-2 ) } = \\ &= (9x^{4}-1)(x^{2}-2) \end{aligned} $$

Step 2 :

Rewrite $ 9x^{4}-1 $ as:

$$ 9x^{4}-1 = (3x^{2})^2 - (1)^2 $$

Now we can apply the difference of squares formula.

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

After putting $ I = 3x^{2} $ and $ II = 1 $ , we have:

$$ 9x^{4}-1 = (3x^{2})^2 - (1)^2 = ( 3x^{2}-1 ) ( 3x^{2}+1 ) $$

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