The term "resultant" can refer to different concepts depending on the context in which it is used. Here are a few common interpretations: 1. **Vector Resultant**: In physics and mathematics, a resultant typically refers to a single vector that is equivalent to the combined effects of two or more vectors. For example, if two forces are acting at an angle to each other, the resultant force can be found using vector addition, which may involve graphical methods or mathematical calculations using trigonometry.
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The resultant turns a shared-root of a polynomial question into a calculation using only coefficients. For nonconstant univariate polynomials over a field , with their actual polynomial degrees , definewhere is the Sylvester matrix. ThenHere is an algebraic closure. The common root need not lie in : two real polynomials may share a nonreal complex root. The subscript identifies which variable is eliminated.[ref]
Lecture 6: Resultants and Discriminants. Algebraic Techniques and Semidefinite Optimization, MIT 6.972, Spring 2006; MIT 6.972, Lecture 6, 28 February 2006, pp. 6-1–6-2, Resultants: Sylvester matrix, root product and common-root criterion
The useful surprise is that a determinant detects a shared root of a polynomial without first finding either list of roots of a polynomial. If the coefficients depend on a parameter, the resultant itself is a polynomial in that parameter. Its zeros pick out possible exceptional parameter values. Elimination theory uses this to reduce a pair of polynomial equations in two variables to a single necessary equation in one variable; eliminating a variable with a resultant gives an exact example below.
A nonzero resultant also supplies a certificate that and are coprime polynomials. The Sylvester matrix then solves for polynomials satisfying , a Bézout's identity for polynomials. Evaluating such an identity at a hypothetical common root would give .