Use synthetic division to find the quotient and remainder when \( 5 x^{4}+9 x^{3}+3 x^{2}+2 x-8 \) (a) Complete this synthetic division table. \( -2) \quad 5 \quad 9 \quad 3 \quad 2 \quad-8 \) (b) Write your answer in the following form: Quotient + Remainder \( x+2 \) \( \frac{5 x^{4}+9 x^{3}+3 x^{2}+2 x-8}{x+2}=\square+\frac{\square}{x+2} \)

Certainly! Let's perform synthetic division on the polynomial \( 5x^{4} + 9x^{3} + 3x^{2} + 2x - 8 \) divided by \( x + 2 \). ### Part (a): Complete the Synthetic Division Table **Step 1:** Identify the root from the divisor \( x + 2 \). Set \( x + 2 = 0 \) ⇒ \( x = -2 \). **Step 2:** Set up the synthetic division table with \( -2 \) on the left and the coefficients of the polynomial on the top row. \[ \begin{array}{c|ccccc} -2 & 5 & 9 & 3 & 2 & -8 \\ \hline & \underline{\quad} & \quad & \quad & \quad & \quad \\ & 5 & \quad & \quad & \quad & \quad \\ \end{array} \] **Step 3:** Perform the synthetic division steps. 1. **Bring down the leading coefficient:** \[ 5 \] 2. **Multiply by \( -2 \) and add to the next coefficient:** \[ 5 \times (-2) = -10 \\ 9 + (-10) = -1 \] 3. **Multiply by \( -2 \) and add to the next coefficient:** \[ -1 \times (-2) = 2 \\ 3 + 2 = 5 \] 4. **Multiply by \( -2 \) and add to the next coefficient:** \[ 5 \times (-2) = -10 \\ 2 + (-10) = -8 \] 5. **Multiply by \( -2 \) and add to the last term:** \[ -8 \times (-2) = 16 \\ -8 + 16 = 8 \] **Completed Synthetic Division Table:** \[ \begin{array}{c|ccccc} -2 & 5 & 9 & 3 & 2 & -8 \\ \hline & & -10 & 2 & -10 & 16 \\ \hline & 5 & -1 & 5 & -8 & 8 \\ \end{array} \] ### Part (b): Express the Result as Quotient Plus Remainder From the synthetic division: - **Quotient:** \( 5x^{3} - x^{2} + 5x - 8 \) - **Remainder:** \( 8 \) Thus, the division can be expressed as: \[ \frac{5x^{4} + 9x^{3} + 3x^{2} + 2x - 8}{x + 2} = 5x^{3} - x^{2} + 5x - 8 + \frac{8}{x + 2} \] **Final Answer:** --- **(a) Completed Synthetic Division Table:** \[ \begin{array}{c|ccccc} -2 & 5 & 9 & 3 & 2 & -8 \\ \hline & & -10 & 2 & -10 & 16 \\ \hline & 5 & -1 & 5 & -8 & 8 \\ \end{array} \] **(b) Division Result:** \[ \frac{5x^{4} + 9x^{3} + 3x^{2} + 2x - 8}{x + 2} = 5x^{3} - x^{2} + 5x - 8 + \frac{8}{x + 2} \] ---