Question

x^4 + x^3 + x^2 + x + 1

How do i factorize this so I can find the roots of this polynomial?

Thanks for checking it out!

Answer 1

You can't factorize that, sorry

Answer 2

Fiddle is probably trying to figure out a way to solve this :P

Answer 3

x^4+x^3+x^2+x+1
=x^2(x^2+x)+(x^2+x)+1
=(x^2+1)(x^2+x)+1

Looking for roots:
(x^2+1)(x^2+x)+1=0
(x^2+1)(x^2+x)=-1

x^2 >=0 --> x^2+1 is always positive
x^2 >= x --> x^2+x >= 0 --> x^2+x is never negative

(always positive number ) * (never negative number) = -1 is a FALSE statement

so no real roots.

Answer 4

*Sigh*

Answer 5

\[x^4+x^3+x^2+x+1=0\quad,\quad x\neq 1\]\[x^4+x^3+x^2+x+1=0\quad/\cdot(x-1)\]\[x^5-1=0\]\[x^5=1=e^{2\pi k}\quad,\quad k=1,2,3,4\]\[x=e^{2\pi k/5}\quad,\quad k=1,2,3,4\]

Answer 6

nikvist that was brilliant!
thank you so much