It is given that A,B,C and D are roots of the quartic equation X^4 - X +2 = 0. Determine, whether or not (AB+CD) is a root of the equation X^3 – 8X –1 = 0.
Finally an exhaustive solution. I found a constant k = ((9 + I*Sqrt[6063])/2)^(1/3)/3^(2/3) + 8/((3*(9 + I*Sqrt[6063]))/2)^(1/3)} which allows me to write the roots of the quartic as
(-Sqrt[-2/Sqrt[k] - k] - Sqrt[k])/2,
(Sqrt[-2/Sqrt[k] - k] - Sqrt[k])/2,
(-Sqrt[2/Sqrt[k] - k] + Sqrt[k])/2,
(Sqrt[2/Sqrt[k] - k] + Sqrt[k])/2
For the quartic q(x) = 2 - x + x^4, these solve to (8 + k^(-1) - k^2)/4 = 0 for the value above, confirming the solution.
For the cubic the 3 distinct perms (ab+cd) give
-1 - 8*k + k^3,
-((-(Sqrt[-2/Sqrt[k] - k]*Sqrt[2/Sqrt[k] - k]) + k)*(-1 - 8*k + k^3))/(2*k),
-((Sqrt[-2/Sqrt[k] - k]*Sqrt[2/Sqrt[k] - k] + k)*(-1 - 8*k + k^3))/(2*k)
Since the common term -1 - 8*k + k^3 = 0, all the perms of the distinct roots solve the cubic.
For all the other (non-distinct) perms it can be verifyied numerically that they do not solve the cubic.
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Posted by goFish
on 2005-12-24 08:39:08 |
Full solution
