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I try to code a method to solve 16th degree polynomial in computer given equation

$a_0x^{16} + a_1x^{15} + a_2x^{14} + ... + a_{16}$

In fact it should be array of $a[n]$ to solve this $a_0x^n + a_1x^{n-1} + a_2x^{n-2} + ... + a_n$ equation

I want to write a function that could solve at least 16 terms. I have read from many source to know that there would not be algebraic solution exist. But are there any solution at all? And can any of it be written in computer code?

Also by coding I don't limit to only float number. It could return multiple complex or quaternion if needed

Thaina
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  • For polynomials of degree 5 or more there are no "algebraic formulas" for solutions. That is a theorem.............Numeric approximations are usually used, – DanielWainfleet Jan 24 '18 at 05:12
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    @DanielWainfleet that's why I mention in the title about "non algebraic solution" – Thaina Jan 24 '18 at 05:16
  • Let $p(z)=b_0+....+b_nz^n$ with $b_n\ne 0\ne b_0$,... For $z\in \Bbb C,$ if $|z|\geq 1+\max {|b_j/.b_n|: j<n}$ or if $|z|\leq (1+\max {|b_j/b_0|: j>0})^{-1}$ then $p(z)\ne 0. $ This narrows down any initial search for zeroes of $p(z).$ Sometimes a change of variable $z=Az'+B $ can improve this..... I may get back to you about your main Q soon, – DanielWainfleet Jan 24 '18 at 05:23
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    I don't limit to only float number. It could return multiple complex or quaternion if needed The reference to quaternions makes no sense in this context. And "float number" is not well-defined math-wise. Also, 16 is not too special as far as polynomials of degree $\ge 5$ go. If you are after numerical approximations to complex roots of arbitrary $16^{th}$ degree polynomials, then state so in the question. – dxiv Jan 24 '18 at 06:28

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