Factoring this thing with 6 degree

[zeion]

Homework Statement

I need to factor this:

u - u^6 -u^3 +1 = 0

Homework Equations

The Attempt at a Solution

I know that (u +1) and (u-1) are roots.. but not sure what to do now without long division..
Do I put them into multiples like (u^3 + 1)(u^3 -1) ??

 

[Dick]

(u^3 + 1)(u^3 -1) is wrong. What's the problem with long division?

 

[zeion]

It takes too long.. I was just wondering if there was simply rule I could follow when it's in this kind of form?

 

[Dick]

It takes too long.. I was just wondering if there was simply rule I could follow when it's in this kind of form?

No, no simple rule. You know (u-1) and (u+1) are factors, but you don't how many times they divide into your original polynomial. In this case, the answer for both is once. You just have to divide them out. If you want to minimize the work a bit you could divide by (u-1)*(u+1)=u^2-1. Saves you dividing by both separately.

 

[zeion]

Ah okay.. so there is no shortcut :(

 

[Mark44]

There is a sort of shortcut, synthetic division. You have to actually do the division, but you don't have to write down all the x powers. You might have learned it already and forgotten it.

 

[zeion]

So I got (u^2-1)(-u^4-u^2-u-1) = -(u+1)(u-1)(u^4+u^2+u+1)
Does that mean there are roots only at u = -1 and u = 1?

 

[Dick]

So I got (u^2-1)(-u^4-u^2-u-1) = -(u+1)(u-1)(u^4+u^2+u+1)
Does that mean there are roots only at u = -1 and u = 1?

The quartic doesn't have any rational roots. It has complex roots, of course. But as far as a factoring exercise, I think you can just leave it there.

 

[zeion]

Oh okay, thanks!