Question

factorize x^5 - x + 1 = 0

i'm not sure how to factor beyond the 3rd/4th degree. how do i do this one?

Answer 1

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

Answer 2

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

Answer 3

\(y=x^5\) is factored

Answer 4

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

Answer 5

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

Answer 6

not according to galois

Answer 7

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

Answer 8

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

Answer 9

factor

Answer 10

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. :(

Answer 11

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

Answer 12

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..

Answer 13

wow man look at the roots!!

http://www.wolframalpha.com/input/?i=factor+x^5-x%2B1%3D0

http://www.wolframalpha.com/input/?i=factor+x^5-x%2B1