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Discriminant (Polynomial)

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

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


The discriminant of the Quadratic Equation

\begin{displaymath}
ax^2+bx+c=0
\end{displaymath} (2)

is usually taken as
\begin{displaymath}
D=b^2-4ac.
\end{displaymath} (3)

However, using the general definition of the Polynomial Discriminant gives
\begin{displaymath}
D\equiv \prod_{i<j} (z_i-z_j)^2={b^2-4ac\over a^2},
\end{displaymath} (4)

where $z_i$ are the Roots.


The discriminant of the Cubic Equation

\begin{displaymath}
z^3+a_2z^2+a_1z+a_0 = 0
\end{displaymath} (5)

is commonly defined as
\begin{displaymath}
D\equiv Q^3+R^2,
\end{displaymath} (6)

where
$\displaystyle Q$ $\textstyle \equiv$ $\displaystyle {3a_1-{a_2}^2\over 9}$ (7)
$\displaystyle R$ $\textstyle \equiv$ $\displaystyle {9a_2a_1-27a_0-2{a_2}^3\over 54}.$ (8)

However, using the general definition of the polynomial discriminant for the standard form Cubic Equation
\begin{displaymath}
z^3+pz=q
\end{displaymath} (9)

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

where $z_i$ are the Roots and
\begin{displaymath}
P=(z_1-z_2)(z_2-z_3)(z_1-z_3).
\end{displaymath} (11)


The discriminant of a Quartic Equation

\begin{displaymath}
x^4+a_3x^3+a_2x^2+a_1x+a_0=0
\end{displaymath} (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.



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© 1996-9 Eric W. Weisstein
1999-05-24