The simplest form of Bézout’s Theorem states that if we have two simultaneous equations, each of which is a polynomial in \(x\) and \(y\), then the number of solutions is at most the product of the degrees of the two polynomials (as long as the two polynomials do not have a non-constant factor in common).
For example, a circle (represented by \((x-a)^2+(y-b)^2=r^2\), a polynomial of degree \(2\)) and a straight line (represented by \(px+qy+r=0\), a polynomial of degree \(1\)) intersect in at most \(2\times1=2\) points, while two conics (each represented by a polynomial of degree \(2\)) intersect in at most \(2\times2=4\) points.
More sophisticated versions of Bézout’s Theorem take into account further possibilities, for example there may be multiple roots (where a line is tangent to a circle, for instance) and there may be complex roots. Even with these included, the number of solutions is still at most the product of the degrees.
