1.11 Continuation of proof

If you studied MATH111 then you will recall that a highest common factor of two polynomials $p(x)$, $q(x)$ is a polynomial of highest degree which divides both $p(x)$ and $q(x)$; the polynomials are coprime if the only common factors are the constants. Assume $g(x)$ has at least one linear factor. Since $a_{s}$ is distinct from $a_{1},\ldots,a_{s-1}$ and is not a root of any of $Q_{1}(x),\ldots Q_{t}(x)$, we deduce that $g_{1}(x)$ and $(x-a_{s})^{m_{s}}$ are coprime. Now we apply Theorem 7.2.7 in MATH111 to see that

for some real polynomials $h(x),k(x)$. Multiplying by ${\frac{f(x)}{g(x)}}$, we obtain