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You are going to win the "most psychic member" award this year... Good call.I suspect the problem has been misquoted, since the given remainder is not correct, whether the "=" should be "+" or "-". I am assuming then that the constant remainders have been reversed, and the linear remainder is in fact $-2x+5$.
By the division algorithm, we may state:
\(\displaystyle P(x)=(x-1)(x-2)Q(x)+R(x)\)
We know the remainder must be a linear function (right?), and so we may state:
(1) \(\displaystyle P(x)=(x-1)(x-2)Q(x)+ax+b\)
And from the remainder theorem we know:
\(\displaystyle P(1)=1\)
\(\displaystyle P(2)=3\)
I suspect the problem has been misquoted, since the given remainder is not correct, whether the "=" should be "+" or "-". I am assuming then that the constant remainders have been reversed, and the linear remainder is in fact $-2x+5$. So, we have instead:
\(\displaystyle P(1)=3\)
\(\displaystyle P(2)=1\)
Using the two equations above and (1), we may get a 2 X 2 linear system in the parameters $a$ and $b$, which will have a unique solution. Can you put all of this together?