se synthetic division to simplify \( \frac{9 x^{3}-25 x^{2}+6}{x-3} \) Vrite your answer in the form \( \mathrm{q}(\mathrm{x})+\frac{\mathrm{r}}{\mathrm{d}(\mathrm{x})} \), where \( \mathrm{q}(\mathrm{x}) \) is a polynomial, r is an nteger, and \( \mathrm{d}(\mathrm{x}) \) is a linear polynomial. Simplify any fractions.

To simplify \( \frac{9x^{3} - 25x^{2} + 6}{x - 3} \) using synthetic division, follow these steps: 1. **Set Up:** - **Divisor root:** \( x - 3 = 0 \) implies \( x = 3 \). - **Coefficients of the dividend:** \( 9 \) (for \( x^3 \)), \( -25 \) (for \( x^2 \)), \( 0 \) (for \( x \), since it's missing), and \( 6 \) (constant term). 2. **Synthetic Division Process:** \[ \begin{array}{c|cccc} 3 & 9 & -25 & 0 & 6 \\ & & 27 & 6 & 18 \\ \hline & 9 & 2 & 6 & 24 \\ \end{array} \] - **Step-by-Step:** - Bring down the **9**. - Multiply by **3**: \( 9 \times 3 = 27 \); add to **-25**: \( -25 + 27 = 2 \). - Multiply by **3**: \( 2 \times 3 = 6 \); add to **0**: \( 0 + 6 = 6 \). - Multiply by **3**: \( 6 \times 3 = 18 \); add to **6**: \( 6 + 18 = 24 \). 3. **Interpret the Result:** - The **quotient** is \( 9x^{2} + 2x + 6 \). - The **remainder** is \( 24 \). 4. **Express the Final Answer:** \[ \frac{9x^{3} - 25x^{2} + 6}{x - 3} = 9x^{2} + 2x + 6 + \frac{24}{x - 3} \] **Final Answer:** \[ 9x^{2} + 2x + 6 + \frac{24}{x - 3} \]