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

To simplify a polynomial, we have to do two things: 1) combine like terms, and 2) rearrange the terms so that they're written in descending order of exponent. First, we combine like terms, which requires us to identify the terms that can be added or subtracted from each other.