# Why is it so hard to find the roots of polynomial equations?

The question that follows was inspired by this question:

When trying to solve for the roots of a polynomial equation, the quadratic formula is much more simple than the cubic formula and the cubic formula is much more simple than the quartic formula.

1. That the general solutions to various polynomial equations are so complex and difficult to derive seems to suggest a fundamental limitation in the problem solving capabilities of the mathematical machinery. Does this intuition of mine make any sense? What should I make of it?
2. Why is it that with each successive degree in a polynomial equation, the solution becomes so much more complex? Can I gain some intuition about what makes finding the roots so hard?
3. According to the Abel-Ruffini theorem: “there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher.” What is so special about the quintic that makes it the cut-off for finding a general algebraic solution?

The idea is basically:

Any monic polynomial can be factored as $f(x) = \prod (x - a_i)$, where $a_{1,\dots,n}$ are the roots of the polynomial.

Now if we expand such a product:

$(x - a_1)(x - a_2) = x^2 - (a_1 + a_2)x + a_1a_2$
$(x - a_1)(x - a_2)(x - a_3) = x^3 - (a_1 + a_2 + a_3)x^2 + (a_1a_2 + a_1a_3 + a_2a_3)x - a_1a_2a_3$

And so on. The pattern should be clear.

This means that finding the roots of a polynomial is in fact equivalent to solving systems like the following:

For a quadratic polynomial $x^2 - px + q$, find $a_1,a_2$, such that

$p = a_1 + a_2$

$q = a_1a_2$

For a cubic polynomial $x^3 - px^2 + qx - r$, find $a_1,a_2,a_3$, such that

$p = a_1 + a_2 + a_3$

$q = a_1a_2 + a_1a_3 + a_2a_3$

$r = a_1 a_2 a_3$

And similarly for higher degree polynomials.

Not surprisingly, the amount of “unfolding” that needs to be done to solve the quadratic system is much less than the amount of “unfolding” needed for the cubic system.

The reason why polynomials of degree 5 or higher are not solvable by radicals, can be thought of as: The structure (symmetries) of the system for such a polynomial just doesn’t match any of the structures that can be obtained by combining the structures of the elementary operations (adding subtracting, multiplication, division, and taking roots).