Discriminant of a polynomial: Difference between revisions

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In algebra, the discriminant of a polynomial is an invariant which determines whether or not a 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 =(-1)^{n(n-1)/2}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 formal 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.

Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 193-194,204-205,325-326. ISBN 0-201-55540-9.

@@ Line 4: / Line 4: @@
 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
+with roots <math>\alpha_1,\ldots,\alpha_n </math>, the discriminant Δ(''f'') with respect to the variable ''x'' is defined as
-:<math>\alpha_1,\ldots,\alpha_n </math>
-the discriminant Δ(''f'') with respect to the variable ''x'' is defined as
 :<math>\Delta = (-1)^{n(n-1)/2} a_n^{2(n-1)} \prod_{i \neq j} (\alpha_i - \alpha_j) . </math>