# How do you simplify polynomials?

You can simplify polynomials only if they have roots. You can think of polynomials as numbers, and of monomials of the form $\left(x - a\right)$ as prime numbers. So, as you can write a composite numbers as product of primes, you can write a "composite" polynomial as product of monomials of the form $\left(x - a\right)$, where $a$ is a root of the polynomial. If the polynomial has no roots, it means that, in a certain sense, it is "prime", and cannot thus be further simplified.
For example, ${x}^{2} + 1$ has no (real) roots, so it cannot be simplified. On the other hand, ${x}^{2} - 1$ has roots $\setminus \pm 1$, so it can be simplified into $x \left(+ 1\right) \left(x - 1\right)$.
Finally, ${x}^{3} + x$ has a root for $x = 0$. So, we can write as $x \left({x}^{2} + 1\right)$, and for what we saw before, this expression is no longer simplifiable.