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In mathematics, a '''resolvent cubic''' [[polynomial]] is defined as follows:
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Let
 
:<math>f(x)=x^4+a_3x^3+a_2x^2+a_1x+a_0 \,</math>
 
be a [[monic polynomial|monic]] [[quartic polynomial]]. The ''resolvent cubic'' is the monic [[cubic polynomial]]
 
:<math>g(x)= x^3+b_2x^2+b_1x+b_0 \,</math>
 
where
 
:<math>b_2 = -a_2 \,</math>
:<math>b_1 = a_1a_3 - 4a_0 \,</math>
:<math>b_0 = 4a_0a_2 - a_1^2 -a_0a_3^2. \,</math>
 
This can be used to solve the quartic, by using the following relations between the roots <math>\alpha_i</math> of ''f'' and the roots <math>\beta_i</math> of ''g'':
:<math>\beta_1 = \alpha_1 \alpha_2 + \alpha_3 \alpha_4</math>
:<math>\beta_2 = \alpha_1 \alpha_3 + \alpha_2 \alpha_4</math>
:<math>\beta_3 = \alpha_1 \alpha_4 + \alpha_2 \alpha_3.</math>
These can be established simply with [[Vieta's formulas]].
 
==See also==
*[[Resolvent (Galois theory)]]
*[[Zero of a function]]
{{clear}}
 
==External references==
*[http://mathworld.wolfram.com/ResolventCubic.html Mathworld reference]
*[http://www.wolframalpha.com/entities/mathworld/resolvent_cubic/rt/uu/ks/ A Wolfram definition]
*[http://www1.wolframalpha.com/input/?i=resolvent+cubic&lk=1&a=ClashPrefs_*MathWorld.ResolventCubic- Wolfram reference]
*[http://www.sosmath.com/algebra/factor/fac12/fac12.html A derivation]
 
[[Category:Polynomials]]

Revision as of 00:41, 11 February 2014



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