Resolvent cubic

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, a resolvent cubic polynomial is defined as follows: Let

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

be a monic quartic polynomial. The resolvent cubic is the monic cubic polynomial

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


b_2 = -a_2 \,
b_1 = a_1a_3 - 4a_0 \,
b_0 = 4a_0a_2 - a_1^2 -a_0a_3^2. \,

This can be used to find the roots of the quartic, by using the following relations between the roots \alpha_i of f and the roots \beta_i of g:

\beta_1 = \alpha_1 \alpha_2 + \alpha_3 \alpha_4
\beta_2 = \alpha_1 \alpha_3 + \alpha_2 \alpha_4
\beta_3 = \alpha_1 \alpha_4 + \alpha_2 \alpha_3.

These can be established simply with Vieta's formulas.

See also[edit]


External links[edit]