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 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)/2a0-1R(f,df/dx).
If you evaluate the determinant you'll obtain the formulae I gave above
in the quadratic and cubic cases.