Partial fraction decomposition in $\mathbb C (X)$ of real fraction

Question

Let $F={P\over Q} $ be a fraction with $deg(P)< deg(Q)$ and such that $P$ and $Q$ are polynomials with real coefficients. Suppose $Q$ has the form $Q=(X-a)(X-z)(X-\bar z)$ with $a\in \mathbb R$ and $z$ is a non real root of $Q$ and $\bar z$ is the conjugate of $z$. The partial fraction decomposition of $F$ has the form $$F={s\over X-a }+{r\over X-z}+{t\over X-\bar z}$$ where $s$ is a real and $r, t$ are complex numbers. I would like to say that $r$ and $t$ must be conjugate but i don't see how to prove it. Thank you for your help!

Answer 1

Because the sum of the second and third term must be a rational function with real coefficients. Write that sum with the real and imaginary parts of r, t and z.