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)=a0xm + a1xm-1 + ... + am 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 + b1xn-1 + ... + bn 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 ... am and you have to read b b ... b as b0 b1 ... bn

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.