Discriminant

discriminant

[di′skrim·ə·nənt]

(mathematics)

The quantity b ²- 4 ac, where a,b,c are coefficients of a given quadratic polynomial: ax ²+ bx + c.

More generally, for the polynomial equation a₀ xⁿ + a₁ x^{n -1}+···+ a_nx₀= 0, a₀^{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) = a₀xⁿ + a₁xⁿ⁻¹ + … + a_n

is the expression

in which the product is distributed over all possible differences of the roots α₁, β₂, … , 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 ax² + bx + c, the discriminant is b² − 4ac. For x³ + px + q, the discriminant is −4p³ −27q². The discriminant differs only by a factor a₀ from the resultant R(P, P′) of the polynomial P(x) and its derivative P′(x).