# Complex zeros of polynomial with no real zeros

bobus
I have to find the complex zeros of the following polynomial:
x^6+x^4+x^3+x^2+1
This evidently doesn't have any real solution so I tried to facto it with long division and by guessing I came up with:
(x^2-x+1)(x^4+x^3+x^2+x+1)

How can I factor the 4th degree polynomial now? Or how can I factor the original polynomial from beginning?

Thanks!!!

## Answers and Replies

Homework Helper
welcome to pf!

hi bobus! welcome to pf! (try using the X2 icon just above the Reply box )
This evidently doesn't have any real solution so I tried to facto it with long division and by guessing I came up with:
(x^2-x+1)(x^4+x^3+x^2+x+1)

How can I factor the 4th degree polynomial now?

that was very good factoring! ok, you have x4 + x3 + x2 + x + 1 …

hmm … let's do a bit of lateral thinking what is it a factor of ? bobus
Ok I'm pretty sure that not even X^4+x^3+x^2+x+1 has any real solution. So I guess there are 4 more complex solutions. So I was thinking about (ax^2+bx+c)(dx^2+ex+f) and trying to come up with some values for a, b, c, d, e and f but it seems to dont work...
Any idea??

Homework Helper
no, you're looking down, look up! what fifth-order expression is it a factor of? bobus
x^5+x^4+x^3+x^2+x
How does this help me?

now subtract something 