What are all possible rational roots for the equation #3x ^ { 4} - 5x ^ { 2} + 25= 0#?

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Answer 1 · 2016-12-22T14:33:50

None. The roots are #=+-1.7078+-i1.4434 #, nearly.

The equation can be reorganized as

#(x^2--5/6)^2=--(5/6sqrt35)^2=i^2(5/6sqrt35)^2# that gives

#x^2=5/6(1+-isqrt35)#. And so,

#x=(5(1/6+-isqrt35/6) )^(1/2)#

#=sqrt5cis((k360^o +-80.406^o)/2), k = 0, 1#, using De Moivre's

theorem

#=sqrt5(cos 40.203^0+-i sin 40.203^0)# and.

#sqrt5(cos 220.203^0+-i sin 220.203^0)#

#=1.7078+-i1.4434 and -1.70755+-i1.4434#

#=+-1.7078+-i1.4434 #