Factor polynomial $$ x^{5}-x^{4}-x+1 $$

$$ x^{5}-x^{4}-x+1 = ( x-1 )^{ 2 }( x+1 )( x^{2}+1 ) $$

Step 1 :

To factor $ x^{5}-x^{4}-x+1 $ we can use factoring by grouping:

Group $ \color{blue}{ x^{5} }$ with $ \color{blue}{ -x^{4} }$ and $ \color{red}{ -x }$ with $ \color{red}{ 1 }$ then factor each group.

$$ \begin{aligned} x^{5}-x^{4}-x+1 = ( \color{blue}{ x^{5}-x^{4} } ) + ( \color{red}{ -x+1 }) &= \\ &= \color{blue}{ x^{4}( x-1 )} + \color{red}{ -1( x-1 ) } = \\ &= (x^{4}-1)(x-1) \end{aligned} $$

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:

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

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