The importance of the fundamental theorem of algebra—known as the d'Alembert–Gauss theorem—is confirmed by the fact that such eminent mathematicians as Lagrange, Euler and d'Alembert, and Gauss, the prince among them, all tackled its proof. It states that every nonconstant polynomial with complex coefficients has at least one root. Consequently, every polynomial of degree n has n complex roots.
The Gauss–Lucas theorem
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Since the derivative of a polynomial of degree n is a polynomial of degree n – 1, it is natural to ask whether the roots of the two polynomials are related. There is a theorem for real-valued functions. If a mountain hike brings you back down to your starting altitude, you must clearly have stopped climbing at some point before beginning your descent. At that point, your route reached a local maximum, where the slope was zero. This is the meaning of Rolle's theorem: if a differentiable real-valued function, such as a polynomial with real coefficients, takes the same value at two points, then its derivative vanishes at least once between them. Thus, if the quadratic polynomial P(x) = ax2 + bx + c has two real roots, its derivative P'(x) = 2a (x + b / 2a) vanishes at x = –b / 2a, half the sum of the roots of P. This observation also holds for complex roots.
A generalization of this property to complex polynomials, used by Gauss as early as 1836 and proved by Félix Lucas in 1874, states that the roots of the derivative P' lie in the convex hull of the roots of the original polynomial P.
Let a complex polynomial be given by
