Describe the Fermat's theorem for polynomials.

Fermat's theorem for polynomials states that every polynomial of degree n has at most n distinct roots.

When a polynomial is factored, each factor corresponds to a root of the polynomial. Fermat's theorem states that the number of distinct roots is at most equal to the degree of the polynomial. This means that a polynomial of degree n can have at most n distinct roots.

To prove Fermat's theorem, we can use mathematical induction. For a polynomial of degree 1, the theorem is true since a linear polynomial has at most one root. Assume that the theorem is true for all polynomials of degree less than or equal to n-1. Now consider a polynomial of degree n. If the polynomial has no roots, then the theorem is trivially true. If the polynomial has a root, then we can factor out the corresponding linear factor. This reduces the degree of the polynomial to n-1, and by the induction hypothesis, the remaining polynomial has at most n-1 distinct roots. Therefore, the original polynomial has at most n distinct roots.

Fermat's theorem has important applications in algebra and calculus. It allows us to determine the maximum number of roots that a polynomial can have, which is useful in solving polynomial equations and in understanding the behavior of functions. For example, if a polynomial has exactly n distinct roots, then it can be written as a product of n linear factors. This can be useful in factorizing polynomials and in finding the roots of a polynomial.