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Solve for 8th degree polynomial?

sanaz ka
Posted 7 years ago

Hello,

Could you please help me to solve this 8th degree polynomial?, I know that according to Abel-Ruffini theorem fifth- and higher-degree equations have no solution in the form of a formula. But, is there any method to find the roots of polynomial with complex coefficient in Mathematica?

Solve[a8*x^8 +a7*x^7 +a6*x^6 + 
  a5*x^5 + a4*x^4 + a3*x^3 + 
 a2*x^2 + a1*x + a0 == 0, x]

Thanks.
POSTED BY: sanaz ka
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sanaz ka
sanaz ka
Posted 7 years ago

Hello Daniel, I need to find x values which can be obtained by solving the roots of the 8 degree polynomial in mathematica. I know that most of these problems are solved by numerical method such as Newtons' method. But is there any general solution for finding the roots of 8th degree polynomial?, (The coefficients of the polynomial are important for my problems).
POSTED BY: sanaz ka

What form of solution are you expecting?
POSTED BY: Daniel Lichtblau
sanaz ka
sanaz ka
Posted 7 years ago

In parametric form help me to put the original numeric value of a0-a8. I wrote that example to show what parametric I mean. In the following, I attach another file to show the value of a0-a8. Anyway, thank you for your help.

Attachments:
POSTED BY: sanaz ka
sanaz ka
sanaz ka
Posted 7 years ago

Many thanks for your response Sander, But all coefficient (a0-a8) have their own numerical values. I need the roots in parametric form. Would you please check the attachment?

Thanks

Attachments:
POSTED BY: sanaz ka

What do you mean with parametric form? You said yourself that can not be as per the Abel-Ruffini theorem.

Your last example in your notebook is a cubic equation in x^2 so that can be solved. But the general case can not be done as you said in your opening post. You can only find the roots by filling in (some of) the variables a0-a8.
POSTED BY: Sander Huisman

Yes it can. Use e.g.:

eq = (x^Range[0, 8]).RandomComplex[{-I - 1, I + 1}, 9]
NSolve[eq==0, x]
POSTED BY: Sander Huisman
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