Use synthetic division to divide the two polynomials. \( \qquad-3 w-5)-9 w^{5}-18 w^{4}+4 w^{3}+21 w^{2}+10 w \) Is the divisor given in \( (x-r) \) form? No What do you need to multiply the divisor by to get a coefficient of 1 ? How many terms are in the dividend? 6 Enter the quotient and remainder. Quotient

To perform synthetic division, we first need to identify the divisor and the dividend. ### Given: - **Dividend**: \( -9w^5 - 18w^4 + 4w^3 + 21w^2 + 10w - 5 \) - **Divisor**: \( -3w - 5 \) ### Step 1: Convert the Divisor The divisor is not in the form \( (w - r) \). To convert it, we can factor out \(-1\) from the divisor: \[ -3w - 5 = -1(3w + 5) \] To make the leading coefficient of the divisor equal to 1, we can multiply the divisor \(3w + 5\) by \(-\frac{1}{3}\): \[ -\frac{1}{3}(3w + 5) = -w - \frac{5}{3} \] ### Step 2: Identify the Number of Terms in the Dividend The dividend \( -9w^5 - 18w^4 + 4w^3 + 21w^2 + 10w - 5 \) has 6 terms. ### Step 3: Perform Synthetic Division We will use synthetic division with the root of the divisor \(3w + 5 = 0\) which gives us \(w = -\frac{5}{3}\). The coefficients of the dividend are: - \( -9 \) (for \( w^5 \)) - \( -18 \) (for \( w^4 \)) - \( 4 \) (for \( w^3 \)) - \( 21 \) (for \( w^2 \)) - \( 10 \) (for \( w^1 \)) - \( -5 \) (constant term) Now we will perform synthetic division using \( -\frac{5}{3} \): 1. Write down the coefficients: \(-9, -18, 4, 21, 10, -5\) 2. Bring down the leading coefficient \(-9\). 3. Multiply \(-9\) by \(-\frac{5}{3}\) and add to the next coefficient. Let's perform the calculations step by step: - Bring down \(-9\). - Multiply: \(-9 \times -\frac{5}{3} = 15\) - Add: \(-18 + 15 = -3\) - Multiply: \(-3 \times -\frac{5}{3} = 5\) - Add: \(4 + 5 = 9\) - Multiply: \(9 \times -\frac{5}{3} = -15\) - Add: \(21 - 15 = 6\) - Multiply: \(6 \times -\frac{5}{3} = -10\) - Add: \(10 - 10 = 0\) - Multiply: \(0 \times -\frac{5}{3} = 0\) - Add: \(-5 + 0 = -5\) ### Final Results The coefficients of the quotient are: - \( -9 \) (for \( w^4 \)) - \( -3 \) (for \( w^3 \)) - \( 9 \) (for \( w^2 \)) - \( 6 \) (for \( w^1 \)) - \( 0 \) (constant term) Thus, the quotient is: \[ -9w^4 - 3w^3 + 9w^2 + 6 \] And the remainder is: \[ -5 \] ### Summary - **Quotient**: \( -9w^4 - 3w^3 + 9w^2 + 6 \) - **Remainder**: \( -5 \)