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) = \(x^2 - x + 1\)