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[Solved] Solutions using Radicals

Anthony Morris, Retired
Posted 4 years ago

Am brand new to Mathematica, but would greatly value your help. Mathematica should be able to give exact roots to any polynomial of degree up to and including four and does so most of the time. However,, when I ask it for the exact solutions to, amongst other examples x^4-x-2=0 it insists on giving me numerical approximations to the equation, whether I use Solve or Reduce. It must be me, what am I doing wrong?
POSTED BY: Anthony Morris
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Thank you, most useful.
POSTED BY: Anthony Morris

Solve will often return Root objects since they are more concise than radicals and also have better numeric behavior. One can influence the choice of return type with the Cubics/Quartics options.

Those Root results are not approximate, by the way. The approximate values shown are from formatting of the Root values. I am not partial to that so I often use InputForm to reformat.
POSTED BY: Daniel Lichtblau
Bill Nelson
Bill Nelson
Posted 4 years ago

Try

ToRadicals[Reduce[x^4-x-2==0,x]]

which returns

x == -1 ||
x == (1 - 2*(2/(47 + 3*Sqrt[249]))^(1/3) + ((47 + 3*Sqrt[249])/2)^(1/3))/3 ||
x == 1/3 + ((1 - I*Sqrt[3])*(2/(47 + 3*Sqrt[249]))^(1/3))/3 - ((1 + I*Sqrt[3])*((47 +  3*Sqrt[249])/2)^(1/3))/6 || 
x == 1/3 + ((1 + I*Sqrt[3])*(2/(47 + 3*Sqrt[249]))^(1/3))/3 - ((1 - I*Sqrt[3])*((47 +  3*Sqrt[249])/2)^(1/3))/6
POSTED BY: Bill Nelson

Thank you very much indeed, this gave me the desired result!
POSTED BY: Anthony Morris
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