Describe the Bézout's theorem for polynomials.

Bézout's theorem states that for two polynomials f(x) and g(x) over a field, there exist polynomials h(x) and k(x) such that h(x)f(x) + k(x)g(x) = d(x), where d(x) is the greatest common divisor of f(x) and g(x).

In other words, Bézout's theorem guarantees the existence of a linear combination of f(x) and g(x) that equals their greatest common divisor. This is useful in finding the roots of polynomials and in polynomial factorisation.

To prove Bézout's theorem, we start by defining the ideal generated by f(x) and g(x) as I = {h(x)f(x) + k(x)g(x) : h(x), k(x) are polynomials}. We can show that I is an ideal of the polynomial ring F[x], where F is the field over which f(x) and g(x) are defined.

Since F[x] is a principal ideal domain, I is generated by a single polynomial d(x), which is the greatest common divisor of f(x) and g(x). Therefore, there exist polynomials h(x) and k(x) such that h(x)f(x) + k(x)g(x) = d(x).

To find h(x) and k(x), we can use the extended Euclidean algorithm. Starting with f(x) and g(x), we perform the Euclidean algorithm to find their greatest common divisor d(x). At each step, we express the remainder as a linear combination of the previous two polynomials, until we reach a remainder of 0. We then work backwards to find h(x) and k(x) by substituting the linear combinations into the previous equations.

For example, let f(x) = x^3 - 3x^2 + 3x - 1 and g(x) = x^2 - 2x + 1. We perform the Euclidean algorithm as follows:

x^3 - 3x^2 + 3x - 1 = (x^2 - 2x + 1)(x - 1) + 0
x^2 - 2x + 1 = (x^3 - 3x^2 + 3x - 1)(x) + (x - 1)

Therefore, d(x) = x - 1, and we can work backwards