Probabilistic relevance model

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

$f(x)=x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}\,$

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

$g(x)=x^{3}+b_{2}x^{2}+b_{1}x+b_{0}\,$

where

$b_{2}=-a_{2}\,$

$b_{1}=a_{1}a_{3}-4a_{0}\,$

$b_{0}=4a_{0}a_{2}-a_{1}^{2}-a_{0}a_{3}^{2}.\,$

This can be used to solve 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

• Resolvent (Galois theory)
• Zero of a function

50 year old Petroleum Engineer Kull from Dawson Creek, spends time with interests such as house brewing, property developers in singapore condo launch and camping. Discovers the beauty in planing a trip to places around the entire world, recently only coming back from .

External references

• Mathworld reference
• A Wolfram definition
• Wolfram reference
• A derivation