Use synthetic division to divide. \[ \frac{7 x^{3}+24 x^{2}+6 x-9}{x+3} \]

To use synthetic division to divide 7x³ + 24x² + 6x - 9 by x + 3, follow these steps: Step 1. Rewrite the divisor in the form x - a. Since the divisor is x + 3, we have a = -3. Step 2. Write the coefficients of the dividend: 7, 24, 6, and -9. Step 3. Set up the synthetic division: -3 | 7 24 6 -9 | ? ? ? ----------------------- ? ? ? ? Step 4. Bring down the first coefficient (7): -3 | 7 24 6 -9 | ----------------------- 7 Step 5. Multiply 7 by -3: 7 × (-3) = -21, and write this under the next coefficient: -3 | 7 24 6 -9 | -21 ----------------------- 7 Step 6. Add the column: 24 + (-21) = 3, and write it below: -3 | 7 24 6 -9 | -21 ----------------------- 7 3 Step 7. Multiply the new value (3) by -3: 3 × (-3) = -9, and place it under the next coefficient: -3 | 7 24 6 -9 | -21 -9 ----------------------- 7 3 Step 8. Add the column: 6 + (-9) = -3, and write it below: -3 | 7 24 6 -9 | -21 -9 ----------------------- 7 3 -3 Step 9. Multiply -3 by -3: (-3) × (-3) = 9, and write it under the last coefficient: -3 | 7 24 6 -9 | -21 -9 9 ----------------------- 7 3 -3 Step 10. Add the column: -9 + 9 = 0, which is the remainder: -3 | 7 24 6 -9 | -21 -9 9 ----------------------- 7 3 -3 0 The bottom row (except for the remainder) gives the coefficients of the quotient polynomial. Since we started with a cubic polynomial, the quotient is a quadratic polynomial: 7x² + 3x - 3. Therefore, the division is: 7x³ + 24x² + 6x - 9 = (x + 3)(7x² + 3x - 3) + 0 Since the remainder is 0, the division is exact. Final Answer: The quotient is 7x² + 3x - 3.