Discriminant (Polynomial)

The Product of the Squares of the differences of the Polynomial Roots $x_i$. For a Polynomial of degree $n$,

$$D_n\equiv \prod_{\scriptstyle i,j\atop\scriptstyle i<j}^n (x_i-x_j)^2.$$ (1)

The discriminant of the Quadratic Equation

$$ax^2+bx+c=0$$ (2)

is usually taken as

$$D=b^2-4ac.$$ (3)

However, using the general definition of the Polynomial Discriminant gives

$$D\equiv \prod_{i<j} (z_i-z_j)^2={b^2-4ac\over a^2},$$ (4)

where $z_i$ are the Roots.

The discriminant of the Cubic Equation

$$z^3+a_2z^2+a_1z+a_0 = 0$$ (5)

is commonly defined as

$$D\equiv Q^3+R^2,$$ (6)

where

$$Q \equiv {3a_1-{a_2}^2\over 9}$$ (7)

$$R \equiv {9a_2a_1-27a_0-2{a_2}^3\over 54}.$$ (8)

However, using the general definition of the polynomial discriminant for the standard form Cubic Equation

$$z^3+pz=q$$ (9)

gives

$$D\equiv \prod_{i<j} (z_i-z_j)^2=P^2=-4p^3-27q^2,$$ (10)

where $z_i$ are the Roots and

$$P=(z_1-z_2)(z_2-z_3)(z_1-z_3).$$ (11)

The discriminant of a Quartic Equation

$$x^4+a_3x^3+a_2x^2+a_1x+a_0=0$$ (12)

is

$-27{a_1}^4+18{a_3}{a_2}{a_1}^3-4{a_3}^3{a_1}^3-4{a_2}^3{a_1}^2+{a_3}^2{a_2}^2{a_1}^2$
$+{a_0}(144{a_2}{a_1}^2-6{a_3}^2{a_1}^2-80{a_3}{a_2}^2{a_1}+18{a_3}^3{a_2}{a_1}+16{a_2}^4$
$-4{a_3}^2{a_2}^3)+{a_0}^2(-192{a_3}{a_1}-128{a_2}^2+144{a_3}^2{a_2}-27{a_3}^4)-256{a_0}^3\quad$	(13)

(Beeler et al. 1972, Item 4).

See also Resultant

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.