Factorize: $x^3 + x + 2$.

Question

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?

Answer 1

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}

Answer 2

Another solution$:$ $$x^3+x+2=(x^3+1)+(x+1)=(x+1)(x^2-x+1)+(x+1)=(x+1)(x^2-x+2)$$

Answer 3

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..