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Question: how to find explicit root of a polynomial of degree6

Question:how to find explicit root of a polynomial of degree6

adel-00 135
April 27 2013
0

need to find the explicit root of u where k,b are  positive constants:

eq5:=u->-3*b*k*u^2-3*b*k^2*u^4-k^3*u^6*b+k*u+5*k^2*u^3-b;

sol:=(solve(eq5(u),u)): S:=array([],1..3): S[1]:=((sol[1]));S[2]:=sol[2]:S[3]:=sol[3]:S[4]:=sol[4]:S[5]:=sol[5]:S[6]:=sol[6]:

and choose the positive real root and substitute it in another equation.

and then plot k against b as:

contourplot(g,k =0.. 20,b = 0 .. 100,contours=[0],axes=boxed,thickness=2,color=black,font=[1,1,18],tickmarks=[5, 4],linestyle=1); where g is the final equation after the substitution of the  real +ve root

The problem that the roots of the above polynomial is of the form:

RootOf(3bk_Z^2+3bk^2_Z^4+k^3_Z^6b-k_Z-5k^2_Z^3+b)
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