Viewing topic: Resolvent Cubic

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Resolvent Cubic

  • Definition: For a given monic quartic polynomial f(x)=x^4 + a_3x^3 + a_2x^2 + a_1x + a_0, the resolvent cubic is the monic cubic polynomial g(x)=x^3 + b_2x^2 + b_1x + b_0, where the coefficients b_i are given in terms of the a_i by b_2 = -a_2, b_1 = a_1a_3 - 4a_0, b_0 = 4a_0a_2 - a_1^2 - a_0a_3^2. 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, beta_2 = alpha_1alpha_3 + alpha_2alpha_4, beta_3 = alpha_1alpha_4 + alpha_2alpha_3. The resolvent cubic of a quartic polynomial 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.

    Source: http://mathworld.wolfram.com

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