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| | In mathematics, a '''resolvent cubic''' [[polynomial]] is defined as follows: |
| | Let |
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| | :<math>f(x)=x^4+a_3x^3+a_2x^2+a_1x+a_0 \,</math> |
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| | be a [[monic polynomial|monic]] [[quartic polynomial]]. The ''resolvent cubic'' is the monic [[cubic polynomial]] |
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| | :<math>g(x)= x^3+b_2x^2+b_1x+b_0 \,</math> |
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| | where |
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| | :<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> |
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| | 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] |
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| | [[Category:Polynomials]] |
In mathematics, a resolvent cubic polynomial is defined as follows:
Let
be a monic quartic polynomial. The resolvent cubic is the monic cubic polynomial
where
This can be used to solve the quartic, by using the following relations between the roots of f and the roots of g:
These can be established simply with Vieta's formulas.
See also
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External references