Factorize: $x^3 + x + 2$. How do I factorize the term $x^3 + x +2$?
What I have previously tried is the middle term factor method but it didn't work...
$x^3 + x + 2$
$\Rightarrow x^3 + 2x - x + 2$
$\Rightarrow x(x^2 + x) - 2(x - 1)$
This doesn't work.
What should I do?
 A: Note
\begin{align}
x^3+x+2& = (x^3+x^2)-(x^2-x-2)\\
&= x^2(x+1)-(x-2)(x+1)\\
&=(x+1)(x^2-x+2)
\end{align}
A: Another solution$:$
$$x^3+x+2=(x^3+1)+(x+1)=(x+1)(x^2-x+1)+(x+1)=(x+1)(x^2-x+2)$$
A: Quanto's answer easily hows how this can be factorised , my answer is another idea of how to find it (you may find it understandable).
Questions like these can't be factorised by Middle Term Factorisation , you can't do it as well as the equation is cubic .
For equations like $x^3 + x + 2$, Try substituting simple values for $x$ like $1,2,3,...$ or $(-1),(-2),(-3),...$, where $x^3 + x + 2 = 0$ .
This may look like a bit of Trial-and-Error , but you can easily find values of $x$ , you can also take some help of Rational Root Theorem to minimize your possible values for $x$ . Rational Root Theorem here says that $x$ can take value of $1,(-1),2,(-2)$ and nothing else.
In this case, $x = (-1)$ works and immediately gives $x^3 + x + 2 = 0$ (you can check it), so you can say that $(x + 1)$ is a factor of $x^3 + x + 2$ , as $(x + 1)$ $= 0$, which gives $x = (-1)$.
Now you can divide $(x^3 + x + 2)$ by $(x + 1)$ to get the other factor (in this case :-  $x^2 - x + 2$)
This way, you see that $(x^3 + x + 2) = (x + 1)(x^2 - x + 2)$
It's easy to see that $(x^2 - x + 2)$ can't be factored..
