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Resolvent Cubic


For a given monic quartic equation

 f(x)=x^4+a_3x^3+a_2x^2+a_1x+a_0,
(1)

the resolvent cubic is the monic cubic polynomial

 g(x)=x^3+b_2x^2+b_1x+b_0,
(2)

where the coefficients b_i are given in terms of the a_i by

b_2=-a_2
(3)
b_1=a_1a_3-4a_0
(4)
b_0=4a_0a_2-a_1^2-a_0a_3^2.
(5)

The roots beta_1, beta_2, and beta_3 of g are given in terms of the roots alpha_1, alpha_2, alpha_3, and alpha_4 of f by

beta_1=alpha_1alpha_2+alpha_3alpha_4
(6)
beta_2=alpha_1alpha_3+alpha_2alpha_4
(7)
beta_3=alpha_1alpha_4+alpha_2alpha_3.
(8)

The resolvent cubic of a quartic equation can be used to solve for the roots of the quartic in terms of the roots of the cubic, which can in turn be solved for using the cubic equation.


See also

Cubic Equation, Quartic Equation

This entry contributed by David Terr

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Cite this as:

Terr, David. "Resolvent Cubic." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ResolventCubic.html

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