Question: A polynomial function that has roots $1$ and $3 + i$. The complex conjugates theorem states that _____ must also be a root.
1. **State the problem:** We are given a polynomial function with roots $1$ and $3 + i$. We need to find the root that must also be included according to the complex conjugates theorem.
2. **Recall the complex conjugates theorem:** This theorem states that if a polynomial has real coefficients and a complex root $a + bi$ (where $b \neq 0$), then its complex conjugate $a - bi$ must also be a root.
3. **Apply the theorem:** Since $3 + i$ is a root, its complex conjugate $3 - i$ must also be a root.
4. **Conclusion:** The root that must also be a root is $3 - i$.
**Final answer:** $3 - i$