Perturbation theory can be applied to many systems, including certain polynomial solutions. Below, we discuss a specific
application to a certain quintic polynomial.

Perturbation theory often takes on a rather scary reputation, since it's generally only called out by name late in one's academic career, and
has a somewhat strong association with particle physics (see Feynman diagrams and Dyson series). However, the essential concepts are ubiquitous
and it's applications are more broad than they may first seem.

Quintic polynomials, famously, have no general formula for their solution. Below, as an example of the simple and powerful concepts of perturbation theory,
we solve a specific quintic polynomial, giving a Taylor series expansion that approximates the solution after truncation.

The Approach

Suppose we'd like to solve the polynomial below for \(x\).

\[
x^5+x-1=0
\]

Of course, we can't apply a general formula for x since there is no quintic equation. However, we may effectively solve for
an expansion of x in terms of some parameter. But what parameter?

\[
x^5+ \epsilon x-1=0
\]

It turns out, our polynomial just so happens to coincide with the above equation when \(\epsilon = 1\). And, since the value of
\(x\) itself now depends on \(\epsilon\), we can assume it has a Taylor expansion in \(\epsilon\) of the form:

We can then substitute this form of \(x(\epsilon)\) into the polynomial (with the parameter), and collect terms according
to order in \(\epsilon\), as below.

Now, since we expect the above to be satisfied regardless of the value of \(\epsilon\), the series above must vanish term by term.
That is, each coefficient of the expansion in \(\epsilon\) above must vanish independently, and hence, we may solve
for the expansion of \(x\), term by term.

Wolfram|Alpha gives a (real) solution of approximately \(~.788\), so we're not too far off!

Further Resources

This example was meant to be short but informative enough to show the general power of perturbative approaches.
Many details have been neglected here such as the choice of placement of the parameter \(\epsilon\), and the
assumption that the power series of \(x\) in \(\epsilon\) converges.