Prove the Rational Roots Theorem: Let $p(x) = a_0 + a_1\,x + \cdots + a_n\,x^n$ be a polynomial with integer coefficients (that is, each $a_j$ is an integer). If $t=r/s$ (where $r$ and $s$ are nonzero integers and $t$ is written in lowest terms, that is, $gcd(|r|,|s|)=1$) is a non-zero rational root of $p(x)$, that is, if $t\ne 0$ and $p(t)=0$, then $r|a_0$ and $s|a_n$.}
Hello, I just wanted to check if my proof is correct, please. Thank you
Proof: We know $p(x)= a_0 +a_1x+\dots+a_nx^n.$ Let $\frac r s$ be a root of $p(x)=0$. Therefore $a_0 +a_1\frac r s+\dots + a_n\frac{r^n}{s^n}=0$
Thus, $a_0s^n +a_1s^{n-1}r+\dots +a_nr^n=0.$ This implies $a_nr^{n-1} + a_{n-1}r^{n-2}s+\dots +a_1s^{n-1}=-\frac {a_0}{r}s^n$.
The left-hand expression is an integer and therefore r is a divisor of $a_0s^n$. Since $r$ and $s$ are prime to each other, r is a divisor of $a_0$. Once again, $\frac {-a_n}{s}r^n = a_0s^{n-1} + a_1s^{n-2}r+\dots +a_{n-1}r^{n-1}$. The right-hand side is an integer and therefore $s$ is a divisor of $a_nr^n$ since r and s are coprime, $s$ is a divisor of $a_n$.