# Discriminant

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## discriminant

[di′skrim·ə·nənt]*b*

^{2}- 4

*ac,*where

*a,b,c*are coefficients of a given quadratic polynomial:

*ax*

^{2}+

*bx*+

*c*.

*a*

_{0}

*x*+

^{n}*a*

_{1}

*x*

^{n}^{-1}+···+

*a*

_{n}x_{0}= 0,

*a*

_{0}

^{2 n-2}times the product of the squares of all the differences of the roots of the equation, taken in pairs.

## Discriminant

The discriminant of a polynomial

*P(x*) = *a*_{0}*x ^{n}* +

*a*

_{1}

*x*

^{n−1}+ … +

*a*

_{n}is the expression

in which the product is distributed over all possible differences of the roots *α*_{1}, *β*_{2}, … , *α _{n}* of the equation

*P (x*) = 0. The discriminant vanishes if and only if there are equal roots among the roots of the polynomial. The discriminant can be expressed through the coefficients of the polynomial

*P(x*) by representing it in the form of a determinant consisting of these coefficients. Thus, for the second-degree polynomial

*ax*

^{2}+

*bx*+

*c*, the discriminant is

*b*

^{2}− 4

*ac*. For

*x*

^{3}+

*px*+

*q*, the discriminant is −4

*p*

^{3}−27

*q*

^{2}. The discriminant differs only by a factor

*a*

_{0}from the resultant

*R(P, P′*) of the polynomial

*P(x*) and its derivative

*P′(x*).