How do I solve an equation like this?

Question

How do I solve following equation for $X$:

$$ AX^n + BX^{n-1} + CX^{n-2} + \dotsb + YX + Z = 0, $$

where $A,B,C,\dotsc,Z,n$ are known?

Answer 1

@Ben S is correct. There are known formulae for $n=1,2,3,4$ and it's known to be impossible to find a general, exact solution beyond this.

However, this does not mean it can't be solved. Numerical methods can be used in general. See for example Newton Raphson methods and other similar things. You will only have at most $n$ real solutions. If $n$ is odd you will have at least 1 real solution.

Another option (common for school problems) is to find one of the solutions by inspection. If $a$ is a solution then $(x-a)$ is a factor, so you can divide the original expression by $(x-a)$ and find another expression in $x^{n-1}$ to solve, which might be easier.

Answer 2

For $n=1,2,3,4$ there are general formulas. For all other $n$, there is no general method for solving such an equation.

Answer 3

I would add more: for $n\ge 5$, Galois theory proves there cannot exist any general formula, but that goes far beyond the level of precalculus.

For $n=5$, if the coefficients that satisfy special relations (more precisely if the Galois group of the equation has certain properties), there exist a formula, or rather a program to calculate the roots, ands the code of this program (in Maple) is several pages long…