Jse synthetic division to simplify \( \frac{x^{3}-19 x+12}{x-4} \) Write 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 the expression \(\frac{x^{3} - 19x + 12}{x - 4}\) using synthetic division, follow these steps: ### Step 1: Set Up the Synthetic Division 1. **Identify the coefficients** of the numerator polynomial \(x^3 - 19x + 12\). Note that the \(x^2\) term is missing, so its coefficient is 0. The coefficients are: \[ 1 \quad 0 \quad -19 \quad 12 \] 2. **Use the root** corresponding to the divisor \(x - 4\), which is \(4\). ### Step 2: Perform Synthetic Division \[ \begin{array}{c|cccc} 4 & 1 & 0 & -19 & 12 \\ \hline & & 4 & 16 & -12 \\ \hline & 1 & 4 & -3 & 0 \\ \end{array} \] - **Bring down the 1**. - **Multiply by 4**: \(1 \times 4 = 4\) and add to the next coefficient: \(0 + 4 = 4\). - **Multiply by 4**: \(4 \times 4 = 16\) and add to the next coefficient: \(-19 + 16 = -3\). - **Multiply by 4**: \(-3 \times 4 = -12\) and add to the last coefficient: \(12 - 12 = 0\). ### Step 3: Write the Result The synthetic division yields: \[ x^{2} + 4x - 3 \quad \text{with a remainder of} \quad 0 \] ### Final Answer \[ \frac{x^{3} - 19x + 12}{x - 4} = x^{2} + 4x - 3 + \frac{0}{x - 4} \] Since the remainder is zero, the simplified form is: \[ x^{2} + 4x - 3 \]