For a given monic quartic equation
(1)
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the resolvent cubic is the monic cubic polynomial
(2)
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where the coefficients are given in terms of the by
(3)
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(4)
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(5)
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The roots , , and of are given in terms of the roots , , , and of by
(6)
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(7)
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(8)
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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.