Factorize $x^4-x^3+x^2-x+1$

Question

I want to factorize $x^4-x^3+x^2-x+1$ in $\Bbb R [x]$ and $\Bbb C [x]$ but I don't know how to do it.

How I can get the complex roots or factorize?

Related : http://math.stackexchange.com/questions/403025/equation-with-high-exponents — lab bhattacharjee, Jun 05 '13 at 14:39

score 5 · Answer 1 · answered Jun 05 '13 at 14:32

5

Let $w=-x$. Note that $(w-1)(w^4+w^2+\cdots +1)=w^5-1$.

answered Jun 05 '13 at 14:32

André Nicolas

score 4 · Answer 2 · answered Jun 05 '13 at 14:34

4

HINT:

From Chapter $XI$ of this and Article $568−570$ of this

the given equation is Reciprocal one, divide either sides by $x^2$

$$\text{so that, }x^2+\frac1{x^2}-\left(x+\frac1x\right)+1=0$$

$$\text{or, }\left(x+\frac1x\right)^2-2-\left(x+\frac1x\right)+1=0$$

Put $x+\frac1x=y$

answered Jun 05 '13 at 14:34

lab bhattacharjee

This is a nicer way of looking at things than the roots of unity approach, if we want algebraic expressions for the coefficients. – André Nicolas Jun 05 '13 at 14:50
I don't always understand what you are getting at, but your answers are almost never what I expect, and I am almost always surprised by them. Thanks. – MJD Jun 05 '13 at 16:43

2 Answers2