Divide. \[ \left(10 x^{3}+13 x^{2}+7 x+6\right) \div(5 x-1) \] Your answer should give the quotient and the remainder. Quotient: Remainder:

We can use polynomial long division to divide $\left(10 x^{3}+13 x^{2}+7 x+6\right)$ by $(5x-1)$: \[ \begin{array}{c|cccc} \multicolumn{2}{r}{2x^{2}+5x+3} & +\frac{9}{5} \\ \cline{2-5} 5x-1 & 10x^{3}+13 x^{2}+7 x+6 & \\ \multicolumn{2}{r}{10x^{3}-2x^{2}} & \\ \cline{2-3} \multicolumn{2}{r}{15x^{2}+7x} & \\ \multicolumn{2}{r}{15x^{2}-3x} & \\ \cline{2-3} \multicolumn{2}{r}{10x+6} & \\ \multicolumn{2}{r}{10x-2} & \\ \cline{2-3} \multicolumn{2}{r}{8} & \\ \end{array} \] Therefore, the quotient is $2x^{2}+5x+3$ and the remainder is $\frac{9}{5}$. We can check our answer by verifying that $(5x-1)(2x^{2}+5x+3)+\frac{9}{5} = 10x^{3}+13 x^{2}+7 x+6$.