Discriminant (thing)

As has already been mentioned the quadratic polynomial ax²+ bx + c has discriminant b² - 4ac. Further the quadratic has repeated roots iff the discriminant vanishes.

Can we generalise this familiar fact to higher degree polynomials? The answer is yes.

For example, a cubic polynomial ax³ + bx² + cx + d has discriminant b²c² - 4ac³ - 4b³d -27a²d² + 18abcd.

In general a polynomial f(x)=a₀x^m + a₁x^m-1 + ... + a_m of degree m has a discriminant D(f) and this discriminant vanishes iff the polynomial has repeated roots.

A formula for the discriminant is easy to give in terms of a certain determinant. To explain this we need to introduce the concept of the resultant of two polynomials. So suppose we have a second polynomial g(x)=b₀xⁿ + b₁x^n-1 + ... + b_n then the resultant R(f,g) of the polynomials is the following m+n x m+n determinant:

 --- m --- --- n ---
    
|a a ... a 0 0 ... 0|   
|0 a ... a a 0 ... 0|   
|...................|   
|0 0 ..... a a ... a|   
|b b ....b 0 0 ... 0|
|0 b ..... b 0 ... 0|
|...................|
|0 0 ..... 0 b ... b|

(here you have to read a a ... a as a₀ a₁ ... a_m and you have to read b b ... b as b₀ b₁ ... b_n

With these preliminaries behind us we can define the discriminant. D(f)=(-1)^m(m-1)/2a₀^-1R(f,df/dx). If you evaluate the determinant you'll obtain the formulae I gave above in the quadratic and cubic cases.