Prove that 7 roots of equation z⁹+z⁶-z⁵+z⁴-z³-1=0 satisfy the condition |z|=1.

Let the roots of the given polynomial (call it P) be z1 thru z9.
Then P=(z-z1)(z-z2)*…(z-z9).
If you start to multiply this out you can see that the z^0 term (=-1) must be equal to (-z1*-z2*…-z9), or, (-1)(z1*z2*…z9).
So, -1=-1 * (product of all the roots), so that product must be 1.

Edited on August 20, 2024, 5:34 pm

Posted by Kenny M on 2024-08-20 17:33:26