The roots (or zeros) of a polynomial $p(x) = 0$ are the values of $x$ that make it true. Finding roots is one of the oldest problems in mathematics — and the story of which polynomials can be solved by formula is the story that leads to Galois theory.

Quadratics: solved for millennia

$$x^2 + bx + c = 0 \implies x = \frac{-b \pm \sqrt{b^2 - 4c}}{2}$$

The discriminant $\Delta = b^2 - 4c$ determines the nature of the roots: - $\Delta > 0$: two distinct real roots - $\Delta = 0$: one repeated root - $\Delta < 0$: two complex conjugate roots

Vieta’s formulas: roots encode coefficients

If $x_1$ and $x_2$ are the roots of $x^2 + bx + c = 0$, then:

$$x_1 + x_2 = -b \qquad x_1 x_2 = c$$

These are the elementary symmetric polynomials — expressions in the roots that are unchanged when you permute the roots. Swapping $x_1$ and $x_2$ leaves both $x_1 + x_2$ and $x_1 x_2$ invariant.

For a cubic $x^3 + bx^2 + cx + d = 0$ with roots $x_1, x_2, x_3$:

$$x_1 + x_2 + x_3 = -b \qquad x_1 x_2 + x_1 x_3 + x_2 x_3 = c \qquad x_1 x_2 x_3 = -d$$

The coefficients of a polynomial are always symmetric functions of its roots. This is a hint that symmetry is fundamental.

Invariants and non-invariants

The expressions $x_1 + x_2$ and $x_1 x_2$ are invariant under all permutations of the roots. But $x_1 - x_2$ is not — swapping gives $x_2 - x_1 = -(x_1 - x_2)$.

However, $(x_1 - x_2)^2$ IS invariant: $(x_1 - x_2)^2 = (x_2 - x_1)^2$. And it equals the discriminant: $(x_1 - x_2)^2 = b^2 - 4c$.

This pattern — expressions becoming invariant under fewer permutations as you distinguish the roots more — is the heart of Galois theory.

Cubics and quartics: solved, but barely

Cubics (degree 3) were solved by del Ferro, Tartaglia, and Cardano in the 1500s. The formula involves square roots nested inside cube roots. Strikingly, for some cubics with three real roots, the formula requires complex intermediate values — the casus irreducibilis.

Quartics (degree 4) were solved by Ferrari shortly after. The formula is even more intricate — nested radicals several layers deep.

Both formulas express the roots using only the four operations ($+, -, \times, \div$) and radicals ($\sqrt[n]{\cdot}$) applied to the coefficients.

The quintic wall

For quintics (degree 5) and beyond, no such formula exists. This was proved by Abel (1824) and Galois (1832). The reason is not that we haven’t been clever enough — it’s that the symmetry structure of the general quintic is fundamentally incompatible with radical expressions.

Understanding why requires the Galois group.