ElI5: Why isn’t there an equation to solve 5-factor polynomials?

Question

There’s the quadratic formula and the one for 3 and 4 factor ones (apologies if I’m wording this wrong), but I just heard that apparently there isn’t anything like a quintic(?) formula and so on. Why is this?? Googling gives me a bunch of confusing terminology that’s difficult to parse.

In: 75

Answer 1

First, let’s clarify some terms. When we talk about “solving” a polynomial, we’re talking about finding its roots. The roots of a polynomial are the values of x that make the polynomial equal to zero. For example, if we have a quadratic polynomial like (x^2 – 3x + 2), the roots are the values of x that make (x^2 – 3x + 2 = 0). In this case, the roots are (x = 1) and (x = 2).

Now, for polynomials of degree 2 (quadratics), 3 (cubics), and 4 (quartics), we have formulas that can find the roots. These are the quadratic formula, Cardano’s formula, and Ferrari’s formula, respectively. These formulas are great because they give us a systematic way to find the roots of any polynomial of degree 2, 3, or 4.

However, for polynomials of degree 5 and higher, things get more complicated. In the 19th century, a mathematician named Évariste Galois proved that there is no general formula, using only the usual algebraic operations (addition, subtraction, multiplication, division, and root extraction), that can find the roots of a fifth-degree polynomial or higher. This is known as the Abel–Ruffini theorem.

The reason for this has to do with the nature of the symmetries of the roots of polynomials. For polynomials of degree 4 and lower, these symmetries form what’s called a “solvable group,” which means that there’s a systematic way to break down the problem of finding the roots into simpler problems. But for degree 5 and higher, the symmetries form a more complex type of group, called a “non-solvable group,” and there’s no way to break down the problem in the same way.

This doesn’t mean that we can’t find the roots of a fifth-degree polynomial at all. It just means that there’s no one-size-fits-all formula that works for all fifth-degree polynomials. We have to use other methods, like numerical approximation, to find the roots in general.