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1.11 Continuation of proof

If you studied MATH111 then you will recall that a highest common factor of two polynomials p(x)p(x), q(x)q(x) is a polynomial of highest degree which divides both p(x)p(x) and q(x)q(x); the polynomials are coprime if the only common factors are the constants. Assume g(x)g(x) has at least one linear factor. Since asa_{s} is distinct from a1,,as-1a_{1},\ldots,a_{s-1} and is not a root of any of Q1(x),Qt(x)Q_{1}(x),\ldots Q_{t}(x), we deduce that g1(x)g_{1}(x) and (x-as)ms(x-a_{s})^{m_{s}} are coprime. Now we apply Theorem 7.2.7 in MATH111 to see that

1=h(x)g1(x)+k(x)(x-as)ms1=h(x)g_{1}(x)+k(x)(x-a_{s})^{m_{s}}

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

f(x)g(x)=f(x)h(x)(x-as)ms+f(x)k(x)g1(x).{\frac{f(x)}{g(x)}}={\frac{f(x)h(x)}{(x-a_{s})^{m_{s}}}}+{\frac{f(x)k(x)}{g_{1% }(x)}}.