Simplify $ z^4+z^2+1$

Question

$$ z^4+z^2+1$$

I'm trying to get this form: $(z^2+z+1)(z^2-z+1)$ but don't know how.

I tried: $z^2(z^2+1)+1$

Answer 1

Hint: Try this instead and see what you can do - $$z^4+z^2+1=(z^4+2z^2+1)-z^2$$

Answer 2

Using just the $\;a^n \pm b^n$ identities:

$$ \require{cancel} z^4+z^2+1=\frac{z^6-1}{z^2-1}=\frac{(z^3-1)(z^3+1)}{z^2-1}=\frac{\cancel{(z-1)}(z^2+z+1)\,\cancel{(z+1)}(z^2-z+1)}{\cancel{z^2-1}} $$

Answer 3

We have $z^4 + 2z^2 - z^2 + 1 = 0$

$z^4 + 2z^2 + 1 - z^2 = 0$

$(z^2 + 1)^2 - z^2 = 0$

$(z^2 + 1 + z)(z^2 + 1 - z)= 0$