A general quintic equation a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0 can be reduced to one of the form y^5 + b_2 y^2 + b_1 y + b_0 = 0, 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.

Bring quintic form | quintic equation | Vieta's formulas