Divide. If the polynomial does not divide evenly, include the remainder as a fraction. \( \left(10 y^{3}-48 y^{2}+54 y\right) \div(5 y-9) \)

To divide the polynomial \(10 y^{3}-48 y^{2}+54 y\) by \(5 y-9\), we can use polynomial long division. 1. Divide the leading term: \(10y^3 \div 5y = 2y^2\). 2. Multiply \(2y^2\) by \(5y - 9\) to get \(10y^3 - 18y^2\). 3. Subtract this result from the original polynomial: \[ (10 y^{3}-48 y^{2}+54 y) - (10y^3 - 18y^2) = -48y^2 + 18y^2 + 54y = -30y^2 + 54y \] 4. Bring down the next term (if necessary) and repeat. Now divide \(-30y^2\) by \(5y\) to get \(-6y\). 5. Multiply \(-6y\) by \(5y - 9\) to get \(-30y^2 + 54y\). 6. Subtract: \[ (-30y^2 + 54y) - (-30y^2 + 54y) = 0 \] 7. Since there's nothing left, we can say the polynomial divides evenly. Therefore, the answer is: \[ 2y^2 - 6y \text{ with a remainder of } 0. \]