A cubic polynomialof variable x is such that p(1)=3,p(2)=5, p(3)=7. Then find the remainder when it is divided by x²-3x+2.

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$$\begin{gathered} \mathrm{Let} \mathrm{the} \mathrm{cubic} \mathrm{polynomial} \mathrm{be} \mathrm{p}\left(\mathrm{x}\right) . \\ \mathrm{Let} \mathrm{the} \mathrm{quotient} \mathrm{be} \mathrm{q}\left(\mathrm{x}\right) . \\ \mathrm{Let} \mathrm{the} \mathrm{remainder} \mathrm{be} \mathrm{ax}+\mathrm{b} . \\ \mathrm{So}, \\ \mathrm{p}\left(\mathrm{x}\right)=\mathrm{q}\left(\mathrm{x}\right) \left({\mathrm{x}}^2-3\mathrm{x}+2\right)+\left(\mathrm{ax}+\mathrm{b}\right) \\ \mathrm{At} \mathrm{x}=1 \\ \mathrm{p}\left(1\right)=\mathrm{q}\left(1\right) \left(1-3+2\right)+\left(\mathrm{a}+\mathrm{b}\right) \\ 3=\mathrm{a}+\mathrm{b} \\ \mathrm{a}+\mathrm{b}=3 ...\left(1\right) \\ \mathrm{At} \mathrm{x}=2 \\ \mathrm{p}\left(2\right)=\mathrm{q}\left(2\right) \left(4-6+2\right) +\left(2\mathrm{a}+\mathrm{b}\right) \\ 5=2\mathrm{a}+\mathrm{b} \\ 2\mathrm{a}+\mathrm{b}=5 ...\left(2\right) \\ \mathrm{Subtracting} \left(1\right) \mathrm{from} \left(2\right) , \mathrm{we} \mathrm{get} \\ \mathrm{a}=2 \\ \mathrm{Substituing} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{a} \mathrm{in} \left(1\right) , \mathrm{we} \mathrm{get} \\ 2+\mathrm{b}=3 \\ \mathrm{b}=1 \\ \mathrm{So} , \mathrm{remainder} \mathrm{is} 2\mathrm{x}+1 . \end{gathered}$$

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