How do I factorise $r^4+r^2+1$?

Question

How do I factorise $r^4+r^2+1$ ? $(r^2+r+1)(r^2-r+1)$ gives $r^4+r^2+1$ But how to split it into these factors? I generally find roots and then write the factors, but $r^4+r^2+1$ seems to have no real root. Thanks!

I found this article which might be of some use: http://mathforum.org/library/drmath/view/73220.html and this post: https://math.stackexchange.com/questions/487060/solving-4th-degreequartic-polynomial-with-all-irrational-roots — turkeyhundt, Apr 20 '18 at 17:33

score 5 · Answer 1 · answered Apr 20 '18 at 17:35

5

$$x^4+x^2+1=\frac{x^6-1}{x^2-1}=\frac{(x^3-1)(x^3+1)}{(x-1)(x+1)} =(x^2+x+1)(x^2-x+1).$$

answered Apr 20 '18 at 17:35

Angina Seng

158,341

score 4 · Accepted Answer · answered Apr 20 '18 at 17:34

4

One may start with $$ r^4+r^2+1=(r^4+2r^2+1)-r^2=(r^2+1)^2-r^2. $$

answered Apr 20 '18 at 17:34

Olivier Oloa

120,989

How do I factorise $r^4+r^2+1$?

2 Answers2