Use synthetic division to find the quotient and remainder when \( -3 x^{3}+10 x^{2}-6 x+9 \) is divided by \( x-3 \) by completing the parts below. (a) Complete this synthetic division table. (b) Write your answer in the following form: Quotient + \( \frac{\text { Remainder }}{x-3} \) \( \frac{-3 x^{3}+10 x^{2}-6 x+9}{x-3}=\square+\frac{\square}{x-3} \)

To perform synthetic division of the polynomial \( -3x^3 + 10x^2 - 6x + 9 \) by \( x - 3 \), we will use the root of the divisor, which is \( 3 \). ### (a) Synthetic Division Table 1. Write down the coefficients of the polynomial: - For \( -3x^3 \), the coefficient is \( -3 \). - For \( 10x^2 \), the coefficient is \( 10 \). - For \( -6x \), the coefficient is \( -6 \). - For the constant term \( 9 \), the coefficient is \( 9 \). So, the coefficients are: \( -3, 10, -6, 9 \). 2. Set up the synthetic division: \[ \begin{array}{r|rrrr} 3 & -3 & 10 & -6 & 9 \\ & & -9 & 3 & -9 \\ \hline & -3 & 1 & -3 & 0 \\ \end{array} \] 3. Perform the synthetic division step-by-step: - Bring down the first coefficient: \( -3 \). - Multiply \( -3 \) by \( 3 \) (the root) to get \( -9 \) and add it to \( 10 \) to get \( 1 \). - Multiply \( 1 \) by \( 3 \) to get \( 3 \) and add it to \( -6 \) to get \( -3 \). - Multiply \( -3 \) by \( 3 \) to get \( -9 \) and add it to \( 9 \) to get \( 0 \). The final row gives us the coefficients of the quotient and the remainder. The quotient is \( -3x^2 + 1x - 3 \) and the remainder is \( 0 \). ### (b) Write the answer in the specified form Since the remainder is \( 0 \), we can write the result of the division as: \[ \frac{-3x^3 + 10x^2 - 6x + 9}{x - 3} = -3x^2 + x - 3 + \frac{0}{x - 3} \] Thus, the final answer is: \[ \frac{-3x^3 + 10x^2 - 6x + 9}{x - 3} = -3x^2 + x - 3 + \frac{0}{x - 3} \] In summary: \[ \frac{-3x^3 + 10x^2 - 6x + 9}{x - 3} = -3x^2 + x - 3 + 0 \]