# What is the general formula for the discriminant of a polynomial of degree n?

##### 1 Answer

See explanation...

#### Explanation:

The discriminant of a polynomial

Given:

#f(x) = a_nx^n+a_(n-1)x^(n-1)+...+a_1x+a_0#

We have:

#f'(x) = na_(n-1)x^(n-1)+(n-1)a_(n-1)x^(n-2)+...+a_1#

The Sylvester matrix of

#((a_4, a_3, a_2, a_1, a_0, 0, 0),(0, a_4, a_3, a_2, a_1, a_0, 0),(0, 0, a_4, a_3, a_2, a_1, a_0),(4a_4, 3a_3, 2a_2, a_1, 0, 0, 0),(0,4a_4,3a_3,2a_2,a_1,0,0),(0, 0, 4a_4, 3a_3, 2a_2, a_1, 0),(0, 0, 0, 4a_4,3a_3,2a_2,a_1))#

Then the discriminant

#Delta = (-1)^(1/2n(n-1))/a_nabs(S_n)#

For

#Delta = (-1)/a_2abs((a_2,a_1,a_0),(2a_2,a_1,0),(0,2a_2,a_1))=a_1^2-4a_2a_0#

(which you might find more recognisable in the form

For

#Delta = (-1)/a_3abs((a_3, a_2, a_1, a_0, 0),(0, a_3, a_2, a_1, a_0),(3a_3, 2a_2, a_1, 0, 0),(0, 3a_3, 2a_2, a_1, 0), (0, 0, 3a_3, 2a_2, a_1))#

#color(white)(Delta) = a_2^2a_1^2-4a_3a_1^3-4a_2^3a_0-27a_3^2a_0^2+18a_3a_2a_1a_0#

The discriminants for quadratics (

The interpretation of the discriminant for higher order polynomials is more limited, but always has the property that the polynomial has repeated zeros if and only if the discriminant is zero.

**Further reading**