Question

If `a` and `k` are real, for what values of `k` does `z^3 +az^2 + kz + ka = 0` have: a) one real root b) 3 real roots?

Answer 1

See below.

Explanation:

`p(z)=z^3 +az^2 + kz + ka = z^2(z+a)+k(z+a)=(z^2+k)(z+a)=0`

so as we can observe, `z^3 +az^2 + kz + ka =0`

always has a real root which is `z = -a`

and if `k < 0` then `p(z)=0` will have three real roots which are

`z = -sqrt k, z=sqrt k, z=-a`