(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Resolve [itex]z^5-1[/itex] into real linear and quadratic factors.

Hence prove that [tex]cos\frac{2\pi}{5}+cos\frac{4\pi}{5}=-\frac{1}{2}[/tex]

2. Relevant equations

[tex]z=cis\theta[/tex]

[tex]z\bar{z}=cis\theta.cis(-\theta)=cos^2\theta+sin^2\theta=1[/tex]

[tex]z+\bar{z}=cis\theta+cis(-\theta)=2cos\theta[/tex]

3. The attempt at a solution

I was able to show that the the roots of [itex]z^5-1=0[/itex] are:

[tex]z=1,cis\frac{2\pi}{5},cis\frac{-2\pi}{5},cis\frac{4\pi}{5},cis\frac{-4\pi}{5}[/tex]

And hence, the real factors are:

[tex](z-1)(z^2-2z.cos\frac{2\pi}{5}+1)(z^2-2z.cos\frac{4\pi}{5}+1)=0[/tex]

But now I'm stuck and not sure how to start proving that last equation.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Complex Numbers ~ Factors

**Physics Forums | Science Articles, Homework Help, Discussion**