Editing Bézout's theorem (section)

===A line and a curve===
As above, one may write the equation of the line in projective coordinates as <math>y=sx+mt.</math> If curve is defined in projective coordinates by a [[homogeneous polynomial]] <math>p(x,y,t)</math> of degree {{mvar|n}}, the substitution of {{mvar|y}} provides a homogeneous polynomial of degree {{mvar|n}} in {{mvar|x}} and {{mvar|t}}. The [[fundamental theorem of algebra]] implies that it can be factored in linear factors. Each factor gives the ratio of the {{mvar|x}} and {{mvar|t}} coordinates of an intersection point, and the multiplicity of the factor is the multiplicity of the intersection point.

If {{mvar|t}} is viewed as the ''coordinate of infinity'', a factor equal to {{mvar|t}} represents an intersection point at infinity.

If at least one partial derivative of the polynomial {{mvar|p}} is not zero at an intersection point, then the tangent of the curve at this point is defined (see {{slink|Algebraic curve|Tangent at a point}}), and the intersection multiplicity is greater than one if and only if the line is tangent to the curve. If all partial derivatives are zero, the intersection point is a [[Algebraic curve#Singular points|singular point]], and the intersection multiplicity is at least two.