Resultant — Definition, Formula & Examples

The resultant is a single number computed from two polynomials that equals zero if and only if the polynomials share a common root. In the context of vectors, the resultant is the single vector obtained by adding two or more vectors together.

Given polynomials $f(x) = a_n x^n + \cdots + a_0$ of degree $n$ and $g(x) = b_m x^m + \cdots + b_0$ of degree $m$ , their resultant $\text{Res}(f, g)$ is the determinant of the $(m+n) \times (m+n)$ Sylvester matrix formed from the coefficients of $f$ and $g$ . The resultant vanishes precisely when $f$ and $g$ have a common factor of positive degree.

Key Formula

\text{Res}(f,g) = \det \begin{pmatrix} a_n & a_{n-1} & \cdots & a_0 & & \\ & a_n & a_{n-1} & \cdots & a_0 & \\ & & \ddots & & & \ddots \\ b_m & b_{m-1} & \cdots & b_0 & & \\ & b_m & b_{m-1} & \cdots & b_0 & \\ & & \ddots & & & \ddots \end{pmatrix}

Where:

$a_n, \ldots, a_0$ = Coefficients of polynomial f(x) of degree n
$b_m, \ldots, b_0$ = Coefficients of polynomial g(x) of degree m

How It Works

For polynomials, you arrange the coefficients of

f

and

g

into the Sylvester matrix: the first

m

rows contain shifted copies of the coefficients of

f

, and the next

n

rows contain shifted copies of the coefficients of

g

. Computing the determinant of this matrix gives the resultant. If

\text{Res}(f, g) = 0

, the two polynomials share at least one common root. If

\text{Res}(f, g) \neq 0

, they have no root in common.

Worked Example

Problem: Determine whether f(x) = x² − 5x + 6 and g(x) = x² − 4x + 4 share a common root by computing their resultant.

Step 1: Both polynomials have degree 2, so the Sylvester matrix is 4×4. Write the coefficients: f has (1, −5, 6) and g has (1, −4, 4).

S = \begin{pmatrix} 1 & -5 & 6 & 0 \\ 0 & 1 & -5 & 6 \\ 1 & -4 & 4 & 0 \\ 0 & 1 & -4 & 4 \end{pmatrix}

Step 2: Compute the determinant of S by cofactor expansion or row reduction.

\det(S) = 0

Step 3: Verify: f(x) = (x−2)(x−3) and g(x) = (x−2)², so they share the root x = 2.

Answer: The resultant is 0, confirming the polynomials share the common root

x = 2

Why It Matters

The resultant lets you detect common roots of two polynomials without actually solving either one. This technique appears in elimination theory, where you remove a variable from a system of polynomial equations, and it is used in computer algebra systems and engineering applications such as finding intersections of curves.

Common Mistakes

Mistake: Confusing the resultant (a scalar from the Sylvester determinant) with the GCD of two polynomials.

Correction: The resultant tells you whether a common factor exists (it equals zero when one does), but it does not tell you what that common factor is. Use the Euclidean algorithm to find the actual GCD.