Polynomials, Cubic Equations and Partial Fractions - Remainder and Factor Theorems

The cubic polynomial \(\mathrm{g}\left( x \right)\) is such that the coefficient of \({{x}^{3}}\) is \(3\) and the roots of the equation \(\mathrm{g}\left( x \right)=0\) are \(1\), \(2\), and \(k\). Given that \(\mathrm{g}\left( x \right)\) has a remainder of \(10\) when divided by \(x-3\), show that \(k=\frac{4}{3}\). Hence or otherwise, find the remainder when \(\mathrm{g}\left( x \right)\) is divided by \(3x-6\).