# 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?

Apr 4, 2017

See below.

#### Explanation:

$p \left(z\right) = {z}^{3} + a {z}^{2} + k z + k a = {z}^{2} \left(z + a\right) + k \left(z + a\right) = \left({z}^{2} + k\right) \left(z + a\right) = 0$

so as we can observe, ${z}^{3} + a {z}^{2} + k z + k a = 0$

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

and if $k < 0$ then $p \left(z\right) = 0$ will have three real roots which are

$z = - \sqrt{k} , z = \sqrt{k} , z = - a$