Resolve into linear and quadratic factors with real coefficients

MATHEMATICS · Subjective · IB Diploma Programme · Mathematics: Analysis and Approaches · Number and Algebra · Number and Algebra

Problem

Resolve

z^5 + 1

into linear and quadratic factors with real coefficients. Hence, or otherwise deduce that,

4 \sin \left( \frac{\pi}{10} \right) \cdot \cos \left( \frac{\pi}{5} \right) = 1.