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Discriminant - Discriminant of a polynomial

Discriminant - Discriminant of a polynomial: Encyclopedia II - Discriminant - 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 discriminant of the polynomial ax2 + bx + c is b2 − 4ac. For the general definition, suppose p(x) = xn + an−1xn−1 + ... ...

Discriminant: Encyclopedia II - Discriminant - Discriminant of a polynomial

Discriminant - 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 discriminant of the polynomial ax² + bx + c is b² − 4ac.

For the general definition, suppose

p(x) = xⁿ + a_n−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     a_n−1     a_n−2      .         .        .    a₀       0           .   .   .   0
 0     1        a_n−1     a_n−2       .        .    .       a₀           0   .   .   0
 0     0        1        a_n−1     a_n−2       .    .       .            a₀  0   .   0
 .     .        .        .         .        .    .
 .     .        .        .         .        .    .
 0     0        0        0         0        1    a_n−1    a_n−2          .   .   .  a₀
 n  (n−1)a_n−1 (n-2)a_n−2   .         .       1a₁   0        0           .   .   .   0
 0     n      (n−1)a_n−1 (n−2)a_n−2   .        .   1a₁       0           0   .   .   0
 0     0        n       (n−1)a_n−1 (n−2)a_n−2  .    .       1a₁          0   0   .   0
 .     .        .        .         .        .    .
 .     .        .        .         .        .    .
 0     0        0        0         0        n  (n−1)a_n−1(n−2)a_n−2      .   .  1a₁  0
 0     0        0        0         0        0    n      (n−1)a_n−1(n−2)a_n−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

Π_{i < j} (r_i − r_j)²

where r₁, ..., r_n are the (complex) numbers such that

p(x) = (x − r₁) (x − r₂) ... (x − r_n)

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 r_i remains valid; the roots have to be taken in some splitting field of the polynomial.

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Algebraic number theory, Conic sections, Discriminant of an algebraic number field, Polynomials, Quadratic forms, algebraic number fields, algebraic number theory, algebraic structures, characteristic, circle, coefficients, complex, conic section, determinant, ellipse, field, fields, hyperbola, mathematics, parabola, polynomial, polynomials, quadratic forms, quadratic formula for the roots, quadratic polynomial, ramification, real, resultant, shape, splitting field, square root, symmetric matrix, up to

Adapted from the Wikipedia article "Discriminant of a polynomial", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki

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