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?

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Answer 1 · 2017-04-04T11:18:04

See below.

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

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

$z = -sqrt k, z=sqrt k, z=-a$

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