math>philosophy 2 years ago factorize x^5 - x + 1 = 0 i'm not sure how to factor beyond the 3rd/4th degree. how do i do this one?

1. satellite73

fairly sure this will not factor over the rationals, since a quick graph shows it has only one real zero

2. math>philosophy

but how does a "quick graph [that] shows it has one zero" show anything? y=x^5 also has one zero as well. not sure what the point is

3. satellite73

$$y=x^5$$ is factored

4. satellite73

if you can find the zeros then you can factor. i am assuming you mean factor over the rational numbers, although maybe i am mistaken about that

5. amistre64

i dont spose theres a simple easy way to "complete a quintic"?

6. satellite73

not according to galois

7. amistre64

you can try newtons method of trial and error to get closer and closer to a real root

8. asnaseer

hmmm - do you really want to factorise this or are you looking for the solutions? if its the solutions you want then use the method suggested by @amistre64

9. math>philosophy

factor

10. asnaseer

the only factorisation I can think of here is as follows:\begin{align} x^5-x+1&=0\\x(x^4-1)+1&=0\\ x(x^2-1)(x^2+1)+1&=0\\ x(x-1)(x+1)(x^2+1)+1&=0\\ \end{align}but I don't think that helps towards getting to the solution. :(

11. cinar

since f(-1)>0 and f(-2)<0 one root is between -1 and -2 by the Bolzano theorem..

12. cinar

since it is odd degree poly. we can say that it has at least one real root.. but I do not know how to say it has only one real root without looking its graph and I also wanna learn it..

13. cinar