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 sum 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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