Solve $$z^3 + 5z^2 + (9 - 5i)z + 10 - 10i = 0$$
I have never dealt with equations with complex numbers in them so this is interesting; first Ill expand.
$$ \implies z^3 + 5z^2- 5iz + 9z + 10 - 10i = 0$$
I am stuck again -_-
any ideas? Thanks!
2026-03-28 11:33:37.1774697617
Solve $z^3 + 5z^2 + (9 - 5i)z + 10 - 10i = 0$
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To find easy solutions with integral real and complex parts (i.e. to find Gaussian integer solutions), we could use a variation of the Rational Root Theorem and try the factors of the linear term $10-10i=10(1-i)$. We try and fail at $\pm 1$, $\pm i$, and $\pm 2$, but we succeed with $2i$.
We can then divide the polynomial on the left hand side of your equation by $z-2i$ and get a quadratic polynomial. The roots of that can be found by the standard quadratic means.