Editing Bézout's theorem (section)

===Using the resultant (plane curves)===
Let {{mvar|P}} and {{mvar|Q}} be two homogeneous polynomials in the indeterminates {{math|''x'', ''y'', ''t''}} of respective degrees {{mvar|p}} and {{mvar|q}}. Their zeros are the [[homogeneous coordinates]] of two [[projective curve]]s. Thus the homogeneous coordinates of their intersection points are the common zeros of {{mvar|P}} and {{mvar|Q}}.

By collecting together the powers of one indeterminate, say {{mvar|y}}, one gets univariate polynomials whose coefficients are homogeneous polynomials in {{mvar|x}} and {{math|t}}.

For technical reasons, one must [[change of coordinates]] in order that the degrees in {{mvar|y}} of {{mvar|P}} and {{mvar|Q}} equal their total degrees ({{mvar|p}} and {{mvar|q}}), and each line passing through two intersection points does not pass through the point {{math|(0, 1, 0)}} (this means that no two point have the same [[Cartesian coordinate system|Cartesian {{mvar|x}}-coordinate]].

The [[resultant]] {{math|''R''(''x'' ,''t'')}} of {{mvar|P}} and {{mvar|Q}} with respect to {{mvar|y}} is a homogeneous polynomial in {{mvar|x}} and {{mvar|t}} that has the following property:  <math>R(\alpha,\tau)=0</math> with <math>(\alpha, \tau)\ne (0,0)</math> if and only if it exist <math>\beta</math> such that <math>(\alpha, \beta, \tau)</math> is a common zero of {{mvar|P}} and {{mvar|Q}} (see {{slink|Resultant|Zeros}}). The above technical condition ensures that <math>\beta</math> is unique. The first above technical condition  means that the degrees used in the definition of the resultant are {{mvar|p}} and {{mvar|q}}; this implies that the degree of {{mvar|R}} is {{mvar|pq}} (see {{slink|Resultant|Homogeneity}}).

As {{mvar|R}} is a homogeneous polynomial in two indeterminates, the [[fundamental theorem of algebra]] implies that {{mvar|R}} is a product of {{mvar|pq}} linear polynomials. If one defines the multiplicity of a common zero of {{mvar|P}} and {{mvar|Q}} as the number of occurrences of the corresponding factor in the product, Bézout's theorem is thus proved.

For proving that the intersection multiplicity that has just been defined equals the definition in terms of a deformation, it suffices to remark that the resultant and thus its linear factors are [[continuous function]]s of the coefficients of {{mvar|P}} and {{mvar|Q}}.

Proving the equality with other definitions of intersection multiplicities relies on the technicalities of these definitions and is therefore outside the scope of this article.