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§4. Resultant of Two Polynomials 61

§4. Resultant of Two Polynomials

The considerations in the section were used in Chapter 2, §2 to determine to what
extent the sets H f (K) determine the form f .
(4.1) Definition. For polynomials f, g in R[x]given by

f (x) = a 0 + a 1 x +· · · + a m x m and g(x) = b 0 + b 1 x +· · · + b n x n

the resultant R( f, g) of f and g is the element of R given by the following (m +
n) × (m + n) determinant.



0 ··· 0
a 0 a 1 ··· a m 



··· 0
0 a 0 ··· a m−1 a m

n rows

............................... 



00 ··· a 0 a 1 ··· ··· a m
R( f, g) = 
b 0 b 1 ··· b n−1 b n 0 ··· 0


0 b 0 ··· b n−1 b n ··· 0 
m rows
...............................




00 ··· b 0 b 1 ··· ··· b n 
We denote the corresponding resultant matrix by [R( f, g)].
(4.2) Theorem. For a factorial ring R and polynomials f, g ∈ R[x], the following
statements are equivalent:
(1) The polynomials f and g have a common factor of strictly positive degree in
R[x].
(2) There exists nonzero polynomials a(x), b(x) in R[x] such that deg a < n =
deg g, deg b < m = deg f , and af + bg = 0.
(3) R( f, g) = 0.
When R( f, g) = 0 there exists polynomials a(x), b(x) in R[x] such that deg a < n,
deg b < m, and R( f, g) = af + bg.
Proof. If u(x) is a common factor of strictly positive degree, then we can write
f (x) = b(x)u(x) and g(x) =−a(x)u(x) where a and b have the desired properties,
so that (1) implies (2). The converse (2) implies (1) follows from factoring of f and
g in R[x] since R[x] is factorial by (4.1).
For any polynomials of the form
a(x) = α 0 + α 1 x +· · · + α n−1 x n−1
and
m−1
b(x) = β 0 + β 1 x +· · · + β m−1 x ,
we see from the formulas for matrix multiplication that