How can I solve this summations math problem involving polynomials?

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The discussion centers on solving the equation kx^n = ∑_{j=1}^n {(j-k)x^{n-j}d_j}, where d_j is a constant dependent on j. The variables k and n are integers, with k constrained between 0 and n but not equal to either. Participants explore whether a general solution for x exists or if a polynomial representation is the best achievable outcome. It is noted that there is no universal formula for solving polynomial equations of all degrees. The conversation emphasizes the complexity of finding exact solutions for such polynomial equations.
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kx^n = \sum_{j=1}^n ({(j-k)x^{n-j}d_j})

dj is a constant but dependent on j, i.e. independent of x. k is an integer varying between n (which is also an integer) and 0 but it can be assumed that k can equal neither n nor 0.

Is there any way of solving this generally for x? Or is a polynomial the best I can do?
 
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You are in fact asking for exact ways of solving polynomial equations.As far as I know,there is no general formula encompassing polynomials of any degree.Take a look at here.
 

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