Discriminant of a polynomial: Difference between revisions

Revision as of 01:31, 18 December 2008

In algebra, the discriminant of a polynomial is an invariant which determines whether or not the polynomial has repeated roots.

Given a polynomial

f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}

with roots

\alpha _{1},\ldots ,\alpha _{n}

the discriminant Δ(f) with respect to the variable x is defined as

\Delta =a_{n}^{2(n-1)}\prod _{i\neq j}(\alpha _{i}-\alpha _{j}).

The discriminant is thus zero if and only if f has a repeated root.

The discriminant may be obtained as the resultant of the polynomial and its derivative.

The discriminant of a quadratic $aX^{2}+bX+c$ is $b^{2}-4ac$ , which plays a key part in the solution of the quadratic equation.

@@ Line 1: / Line 1: @@
 In [[algebra]], the '''discriminant of a polynomial''' is an invariant which determines whether or not the polynomial has repeated roots.
-The discriminant may be defined as the [[resultant (algebra)|resultant]] of the polynomial and its [[derivative]].
+Given a polynomial
+:<math>f(x)= a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 </math>
+with roots
+:<math>\alpha_1,\ldots,\alpha_n </math>
+the discriminant Δ(''f'') with respect to the variable ''x'' is defined as
+:<math>\Delta = a_n^{2(n-1)} \prod_{i \neq j} (\alpha_i - \alpha_j) . </math>
+The discriminant is thus zero if and only if ''f'' has a repeated root.
+The discriminant may be obtained as the [[resultant (algebra)|resultant]] of the polynomial and its [[derivative]].
 ==Examples==
 The discriminant of a quadratic <math>aX^2 + bX + c</math> is <math>b^2 - 4ac</math>, which plays a key part in the solution of the [[quadratic equation]].