Bézout's theorem for curves states that, in general, two algebraic curves of degrees mand n intersect in points and cannot meet in more than points unless they have a component in common (i.e., the equations defining them have a common factor; Coolidge 1959, p. 10).
Bézout's theorem for polynomials states that if p and Q are two polynomials with no roots in common, then there exist two other polynomials A and B such that . Similarly, given N polynomial equations of degrees ,, ... in N variables, there are in general common solutions.
Séroul (2000, p. 10) uses the term Bézout's theorem for the following two theorems.
1. Let be any two integers, then there exist such that

2. Two integers a and b are relatively prime if there exist such that