MHB What are the roots of this polynomial with a beta coefficient?

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The polynomial in question is $\beta m^5 + m^2 + 1 = 0$. Participants discuss the challenges of finding its roots, particularly due to the presence of the beta coefficient. It is noted that the polynomial is unlikely to factor, complicating the solution process. Ultimately, one contributor indicates they no longer need to solve the polynomial. The discussion highlights the complexities introduced by the beta coefficient in polynomial equations.
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$\beta m^5 + m^2 + 1 =0$

How do I find the roots?
 
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Could you factor? The beta is throwing me a tad, and I imagine that is some of the difficulty of this problem.
 
alane1994 said:
Could you factor? The beta is throwing me a tad, and I imagine that is some of the difficulty of this problem.

It isn't going to factor but I don't need to solve it anymore.
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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