As has already been mentioned the

quadratic polynomial

*ax*^{2}+ bx + c has discriminant

*b*^{2} - 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 *ax*^{3} + bx^{2} + cx + d
has discriminant
*b*^{2}c^{2} - 4ac^{3} - 4b^{3}d -27a^{2}d^{2} + 18abcd.

In general a polynomial
*f(x)=a*_{0}x^{m} + a_{1}x^{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)=b*_{0}x^{n} + b_{1}x^{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 a

_{0} a

_{1} ... a

_{m}
and you have to read b b ... b as b

_{0} b

_{1} ... b

_{n}
With these preliminaries behind us we can define the discriminant.
*D(f)=(-1)*^{m(m-1)/2}a_{0}^{-1}R(f,df/dx).
If you evaluate the determinant you'll obtain the formulae I gave above
in the quadratic and cubic cases.