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

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

Step 1 :

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

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

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

Step 2 :

To factor $ x^{3}-1 $ we can use difference of cubes formula:

$$ I^3 - II^3 = (I - II)(I^2 + I \cdot II + II^2) $$

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

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

Step 3 :

Step 3: Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$. ( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:

$$ \color{blue}{ b = 1 } ~ \text{ and } ~ \color{red}{ c = 1 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 1 } $ and multiply to $ \color{red}{ 1 } $.

Step 4: Find out pairs of numbers with a product of $\color{red}{ c = 1 }$.

PRODUCT = 1
1     1	-1     -1

Step 5: Because none of these pairs will give us a sum of $ \color{blue}{ 1 }$, we conclude the polynomial cannot be factored.

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