Answer
$-3$ and $5i$
Work Step by Step
The Conjugate Pairs Theorem states that when a polynomial has real coefficients, then any complex zeros occur in conjugate pairs. This means that, when $(p +i \ q)$ is a zero of a polynomial function with a real number of the coefficients, then its conjugate $(p –i q)$, is also a zero of the function.
Since we are told that $-5i$ is one zero, the other zero must be $5i$ (the conjugate of $−5i$).
The third zero must be a real number that can be computed by factoring the polynomial:
$f(x) = x^3+3x^2+25x-+75 \\ = (x^3+3x^2)+(25x+75) \\= (x+3)(x^2+25)$
By the zero product property, we have: $x=-3$ and $x^2=-25$. Thus, the real zero of the function is $-3$. This implies that the remaining zeros are: $-3$ and $5i$
