Discriminant of a polynomialThe discriminant of a polynomial is a number which can be easily computed from the coefficients of the polynomial and which is zero if and only if the polynomial has a multiple root. For instance, the discrimant of the polynomial ax2 + bx + c is b2 - 4ac.
For the general definition, suppose
- p(x) = xn + an-1xn-1 + ... + a1x + a0
1 an-1 an-2 . . . a0 0 . . . 0 0 1 an-1 an-2 . . . a0 0 . . 0 0 0 1 an-1 an-2 . . . a0 0 . 0 . . . . . . . . . . . . . . 0 0 0 0 0 1 an-1 an-2 . . . a0 n (n-1)an-1 (n-2)an-2 . . 1a1 0 0 . . . 0 0 n (n-1)an-1 (n-2)an-2 . . 1a1 0 0 . . 0 0 0 n (n-1)an-1 (n-2)an-2 . . 1a1 0 0 . 0 . . . . . . . . . . . . . . 0 0 0 0 0 n (n-1)an-1 an-2 . . 1a1 0 0 0 0 0 0 0 n (n-1)an-1 an-2 . . 1a1
In the case n=4, this discriminant looks like this:
The discriminant of p(x) is thus equal to the resultant of p(x) and p'(x).
One can show that, up to sign, the discriminant is equal to
- Πi<j (ri - rj)2
- p(x) = (x - r1) (x - r2) ... (x - rn)
In order to compute discriminants, one does not evaluate the above determinant each time for different coefficient, but instead one evaluates it only once for general coefficients to get an easy-to-use formula. For instance, the discriminant of a polynomial of third degree is a12a22 - 4a0a23 -4a13 + 18 a0a1a2 - 27a02.
The discriminant can be defined for polynomials over arbitrary fields, in exactly the same fashion as above. The product formula involving the roots ri remains valid; the roots have to be taken in some splitting field of the polynomial.