The Fundamental Theorem of Algebra

In the 16th century Cardano noted that the sum of the three solutions to a cubic equation

x ³ + bx ² + cx + d = 0

is – b, the negation of the coefficient of x ². By the 17th century the theory of equations had developed so far as to allow Girard (1595-1632) to state a principle of algebra, what we call now " the fundamental theorem of algebra". His formulation, which he didn't prove, also gives a general relation between the n solutions to an n ^th degree equation and its n coefficients.

An n ^th degree equation can be written in modern notation as

xⁿ + a ₁ x^{n –1} +... + a_{n –2}x² + a_{n –1} x + a_n = 0

where the coefficients a ₁,..., a_{n –2}, a_{n –1}, and a_n are all constants. Girard said that an n ^th degree equation admits of n solutions, if you allow all roots and count roots with multiplicity. So, for example, the equation x ² + 1 = 0 has the two solutions

√ –1 and –√ –1, and the equation x ² – 2 x + 1 = 0 has the two solutions 1 and 1. Girard wasn't particularly clear what form his solutions were to have, just that there be n of them: x ₁, x ₂,..., x_{n –1}, and x_n.

Girard gave the relation between the n roots x ₁, x ₂,..., x_n, and x_n and the n coefficients a ₁,..., a_{n –2}, a_{n –1}, and a_n that extends Cardano's remark. First, the sum of the roots x ₁ + x ₂ +..., + x_n is – a ₁, the negation of the coefficient of x^{n –1} (Cardano's remark). Next, the sum of all products of pairs of solutions is a ₂. Next, the sum of all products of triples of solutions is – a ₃. And so on until the product of all n solutions is either a_n (when n is even) or – a_n (when n is odd).

Here's an example. The 4^th degree equation

x ⁴ – 6 x ³ + 3 x ² + 26 x – 24 = 0

has the four solutions –2, 1, 3, and 4. The sum of the solutions equals 6, that is –2 + 1 + 3 + 4 = 6. The sum of all products of pairs (six of them) is

(–2)(1) + (–2)(3) + (–2)(4) + (1)(3) + (1)(4) + (3)(4)

which is 3. The sum of all products of triples (four of them) is

(–2)(1)(3) + (–2)(1)(4) + (–2)(3)(4) + (1)(3)(4)

which is 26. And the product of all four solutions is –24.

Descartes (1596–1650) also studied this relation between solutions and coefficients, and showed more explicitly why the relationship holds. Descartes called negative solutions " false" and treated other solutions (that is, complex numbers) " imaginary".

Over the remainder of the 17th century, negative numbers rose in status to be full–fledged numbers. But complex numbers remained in limbo through most of the 18th century. They weren't considered to be real numbers, but they were useful in the theory of equations. It wasn't even clear what form the solutions to equations might take. Certainly complex numbers of the form a + b √ –1 were sufficient to solve quadratic equations, but it wasn't clear whether they were enough to solve cubic and higher-degree equations. Also, the part of the Fundamental Theorem of Algebra which stated there actually are n solutions of an n th degree equation was yet to be proved, pending, of course, some description of the possible forms that the solutions might take.