The complex number $z$ is chosen so that $1, z, z^2, eq ext{...}, z^7$ form the vertices of the regular polygon shown - HSC - SSCE Mathematics Extension 2 - Question 1 - 2017 - Paper 1

To find the polynomial equation that has the complex numbers $1, z, z^2, ext{...}, z^7$ as roots, we note that these roots are the 7th roots of unity. The general form for finding roots of unity is given by the equation:

$z^n - 1 = 0$

For the case of 7 roots, we have:

$z^7 - 1 = 0$

The correct answer is Option A: $z^7 - 1 = 0$. This polynomial has the required complex numbers as its roots.