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# On dividing $$x^3 - 3x^2 + x + 2$$ by a polynomial g(x), the quotient and remainder were $$x-2$$ and $$-2x+4$$ respectively. Find g(x).

Here, p(x) = $$x^3 - 3x^2 + x + 2$$,
q(x) = $$x-2$$,
r(x) = $$-2x+4$$

=>According to polynomial division algorithm,

=>p(x) = g(x).q(x) + r(x)
=>$$x^3 - 3x^2 + x + 2$$ = g(x).$$(x-2)$$ + $$(-2x+4)$$
=> $$x^3 - 3x^2 + x + 2 + 2x - 4$$ = g(x).$$(x-2)$$
=> $$x^3 - 3x^2 + 3x - 2$$ = g(x).$$(x-2)$$
=> $$g(x) = {{x^3 - 3x^2 + 3x - 2}\over{(x-2)}}$$

Solving it using long division:

$$\begin{array}{rrrr|ll} x^3 & -3x^2 & +3x & -2 & x - 2 \\ -x^3 & +2x^2 & & & x^2 - x + 1 \\ \hline & -x^2 & +3x & -2\\ \phantom{-} & +x^2 & -2x & & & \\ \hline & & x & -2 \\ & & -x & + 2 \\ \hline & & & 0\\ \hline \end{array}$$

=>g(x) = $$x^2 - x + 1$$