Answer
The fourth zero must be a real number because complex zeros come in pairs by the Conjugate Pairs Theorem. If we had another complex zero, it would come in a pair, and we would have five zeros total (a contradiction).
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 real number coefficients, then its conjugate $(p –i q)$, is also a zero of the function.
We see that the polynomial function has a degree of $4$, so it has $4$ zeros. We have $2$,$1+2i$, and $1−2i$ as zeros. This implies that the fourth zero must be a real number because complex zeros come in pairs by the Conjugate Pairs Theorem. If we had another complex zero, it would come in a pair, and we would have five zeros total (a contradiction).
