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Discriminant

lastmod 2023-01-12created 2014-10-299 edits

In algebra, the discriminant (判別式) is a relation among the coefficients of a quadratic equation that gives information about the nature of its roots. Symbols such as \(D\) and \(\Delta\) are commonly used. The discriminant of a univariate polynomial \(f\) can be derived from the resultant of \(f\) and the derivative of \(f\).

Quadratic equation

For example, the discriminant of the following quadratic equation

\[ ax^2 + bx + c=0 \] is

\[ D = b^2 - 4ac \] .

Here, when \(a, b, c\) are real numbers,

  • if \(D>0\), the quadratic equation has two distinct real roots,
  • if \(D=0\), the quadratic equation has a repeated real root (double root), and
  • if \(D<0\), the solutions of the quadratic equation are two distinct imaginary roots.

These facts can be determined.

Also, the discriminant of the following cubic equation

\[ ax^3 + bx^2 + cx + d=0 \] is

\[ D = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd \] . Also, the discriminant of the following quartic equation

\[ ax^4 + bx^3 + cx^2 + dx+e=0 \] is

\[ D= b^2 c^2 d^2 -b^3 4d^3 -a c^3 4d^2+abc18d^3-a^2 27d^4 +a^3 256e^3 \] \(-b^2 4c^3+b^3 c 18d e+a 16c^4 e - a b c^2 80d e-a b^2 6d^2 e+a^2 c 144d^2 e\)\(-27b^4 e^2+a b^2 144c e^2 -a^2 128c^2 e^2-a^2 b 192d e^2\)

which comes out to 16 terms.

For higher-degree polynomials as well, the discriminant is always a relation among the coefficients of the equations. These get quite long. The discriminants of degree-5 and degree-6 polynomials consist of 59 and 246 terms, respectively. And the number of terms in the discriminant grows exponentially as the degree of the polynomial increases. For an equation of any degree, having a repeated root over the complex numbers is equivalent to the discriminant being \(0\). This concept applies when the coefficients of the polynomial are real numbers. In this case, the vanishing of the discriminant is equivalent to the polynomial having multiple roots in its splitting field.

See also

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