Find the quotient and remainder using synthetic division. \[ \frac{3 x^{4}+8 x^{3}+5 x^{2}+x+1}{x+\frac{2}{3}} \] quotient remainder

To perform synthetic division with the polynomial \(3x^4 + 8x^3 + 5x^2 + x + 1\) divided by \(x + \frac{2}{3}\), we first convert the divisor to the form \(x - r\) where \(r = -\frac{2}{3}\). ### Step 1: Set Up Synthetic Division The coefficients of the polynomial are: - \(3\) (for \(x^4\)) - \(8\) (for \(x^3\)) - \(5\) (for \(x^2\)) - \(1\) (for \(x^1\)) - \(1\) (constant term) We'll use \(-\frac{2}{3}\) as our synthetic divisor. ### Step 2: Synthetic Division Process 1. Write the coefficients: \(3, 8, 5, 1, 1\). 2. Bring down the leading coefficient \(3\) down to the bottom row. 3. Multiply \(3\) by \(-\frac{2}{3}\), which gives \(-2\), and add this to the next coefficient \(8\), resulting in \(6\). 4. Repeat the process: - Multiply \(6\) by \(-\frac{2}{3}\), giving \(-4\), add to \(5\) to get \(1\). - Multiply \(1\) by \(-\frac{2}{3}\), giving \(-\frac{2}{3}\), add to \(1\) to get \(\frac{1}{3}\). The process looks like this: \[ \begin{array}{r|rrrrr} -\frac{2}{3} & 3 & 8 & 5 & 1 & 1 \\ & & -2 & -4 & -\frac{2}{3} & -\frac{2}{9} \\ \hline & 3 & 6 & 1 & \frac{1}{3} & \frac{1}{3} \\ \end{array} \] ### Step 3: Determining the Quotient and Remainder The bottom row gives us the coefficients of the quotient and the final value is the remainder: - Quotient: \(3x^3 + 6x^2 + 1x + \frac{1}{3}\) - Remainder: \(\frac{1}{3}\) ### Final Output Quotient: \(3x^3 + 6x^2 + x + \frac{1}{3}\) Remainder: \(\frac{1}{3}\)