Use synthetic division to find the result when \( x^{3}-9 x^{2}+4 x+4 \) is divided by \( x-1 \). If there is a remainder, express the result in the form \( q(x)+\frac{r(x)}{b(x)} \)

To divide \( x^{3} - 9x^{2} + 4x + 4 \) by \( x - 1 \) using synthetic division, follow these steps: ### Step 1: Set Up Synthetic Division - **Divisor**: \( x - 1 \) implies \( c = 1 \). - **Coefficients of the Dividend**: \( 1 \) (for \( x^3 \)), \( -9 \) (for \( x^2 \)), \( 4 \) (for \( x \)), and \( 4 \) (constant term). ### Step 2: Perform Synthetic Division \[ \begin{array}{c|cccc} 1 & 1 & -9 & 4 & 4 \\ \hline & 1 & -8 & -4 & 0 \\ \end{array} \] **Process:** 1. Bring down the first coefficient: **1**. 2. Multiply by \( c = 1 \): \( 1 \times 1 = 1 \). 3. Add to the next coefficient: \( -9 + 1 = -8 \). 4. Multiply by \( c = 1 \): \( -8 \times 1 = -8 \). 5. Add to the next coefficient: \( 4 + (-8) = -4 \). 6. Multiply by \( c = 1 \): \( -4 \times 1 = -4 \). 7. Add to the last coefficient: \( 4 + (-4) = 0 \). ### Step 3: Interpret the Result - **Quotient**: The numbers on the bottom row (excluding the remainder) represent the coefficients of the quotient polynomial. Thus, the quotient is: \[ x^2 - 8x - 4 \] - **Remainder**: The last number is \( 0 \), indicating that there is no remainder. ### Final Answer When \( x^{3} - 9x^{2} + 4x + 4 \) is divided by \( x - 1 \), the result is: \[ x^{2} - 8x - 4 \] There is no remainder.