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Principal Quintic Form


A general quintic equation

 a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0=0
(1)

can be reduced to one of the form

 y^5+b_2y^2+b_1y+b_0=0,
(2)

called the principal quintic form.

Vieta's formulas for the roots y_j in terms of the b_js is a linear system in the b_j, and solving for the b_js expresses them in terms of the power sums s_n(y_j). These power sums can be expressed in terms of the a_js, so the b_js can be expressed in terms of the a_js. For a quintic to have no quartic or cubic term, the sums of the roots and the sums of the squares of the roots vanish, so

s_1(y_j)=0
(3)
s_2(y_j)=0.
(4)

Assume that the roots y_j of the new quintic are related to the roots x_j of the original quintic by

 y_j=x_j^2+alphax_j+beta.
(5)

Substituting this into (1) then yields two equations for alpha and beta which can be multiplied out, simplified by using Vieta's formulas for the power sums in the x_j, and finally solved. Therefore, alpha and beta can be expressed using radicals in terms of the coefficients a_j. Again by substitution into (◇), we can calculate s_3(y_j), s_4(y_j) and s_5(y_j) in terms of alpha and beta and the x_j. By the previous solution for alpha and beta and again by using Vieta's formulas for the power sums in the x_j, we can ultimately express these power sums in terms of the a_j.


See also

Bring Quintic Form, Quintic Equation, Vieta's Formulas

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Cite this as:

Weisstein, Eric W. "Principal Quintic Form." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrincipalQuinticForm.html

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