How do you find all the real and complex roots of $x^7 – x^5 + x^2 – 3x + 3 = 0$ ?

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Answer 1 · 2016-07-22T06:32:13

Use a numerical method to find approximations:

$x ~~ -1.50525$

$x ~~ 0.443441+-0.934112i$

$x ~~ -0.740689 +- 1.01346i$

$x ~~ 1.04988+-0.284124i$

In common with most polynomials of degree $5$ or greater, this septic equation has no solutions expressible using $n$ th roots.

About the best we can do is find numerical approximations to the roots using a method like Durand-Kerner.

See an example for a quintic, describing the method at: https://socratic.org/s/awnFhaqT

Here's an example C++ program that implements the method for this septic:

enter image source here

Using this program I found the following approximations:

$x ~~ -1.50525$

$x ~~ 0.443441+-0.934112i$

$x ~~ -0.740689 +- 1.01346i$

$x ~~ 1.04988+-0.284124i$

How do you find all the real and complex roots of x^7 – x^5 + x^2 – 3x + 3 = 0x^7 – x^5 + x^2 – 3x + 3 = 0?