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# Discriminant of a polynomial

The discriminant of a polynomial is a number which can be easily computed from the coefficients of the polynomial and which is zero if and only if the polynomial has a multiple root. For instance, the discrimant of the polynomial ax2 + bx + c is b2 - 4ac.

For the general definition, suppose

p(x) = xn + an-1xn-1 + ... + a1x + a0
is a polynomial with real coefficients. The discriminant of this polynomial is defined as the determinant of the (2n-1) × (2n-1) matrix

``` 1     an-1     an-2      .         .        .    a0       0        .   .   .   0
0     1        an-1     an-2       .        .    .       a0        0   .   .   0
0     0        1        an-1     an-2       .    .       .        a0   0   .   0
.     .        .        .         .        .    .
.     .        .        .         .        .    .
0     0        0        0         0        1    an-1    an-2       .   .   .  a0
n  (n-1)an-1 (n-2)an-2   .         .       1a1   0        0        .   .   .   0
0     n      (n-1)an-1 (n-2)an-2   .        .   1a1       0        0   .   .   0
0     0        n       (n-1)an-1 (n-2)an-2  .    .       1a1       0   0   .   0
.     .        .        .         .        .    .
.     .        .        .         .        .    .
0     0        0        0         0        n  (n-1)an-1  an-2      .   .  1a1  0
0     0        0        0         0        0    n      (n-1)an-1  an-2 .   .  1a1
```

In the case n=4, this discriminant looks like this:

The discriminant of p(x) is thus equal to the resultant of p(x) and p'(x).

One can show that, up to sign, the discriminant is equal to

Πi<j (ri - rj)2
where r1, ..., rn are the (complex) numbers such that
p(x) = (x - r1) (x - r2) ... (x - rn)
Therefore, p has a multiple root if and only if the discriminant is zero. Note however that this multiple root can be complex.

In order to compute discriminants, one does not evaluate the above determinant each time for different coefficient, but instead one evaluates it only once for general coefficients to get an easy-to-use formula. For instance, the discriminant of a polynomial of third degree is a12a22 - 4a0a23 -4a13 + 18 a0a1a2 - 27a02.

The discriminant can be defined for polynomials over arbitrary fields, in exactly the same fashion as above. The product formula involving the roots ri remains valid; the roots have to be taken in some splitting field of the polynomial.

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