Final answer:
To use synthetic division, divide the polynomial x³ - 11x² + 31x - 21 by (x - 3). The result, or quotient polynomial, is x³ - 33x² - 54x - 75.
Step-by-step explanation:
To use synthetic division to determine the polynomial, we will divide the polynomial by the given divisor, which is (x - 3).
Step 1: Write the coefficients of the polynomial in descending order of exponents:
1 | -11 | 31 | -21
Step 2: Bring down the coefficient of the first term, which is 1.
Step 3: Multiply the divisor, 3, by the value you brought down, 1. The result is 3. Write this below the next coefficient. Add this value to the next coefficient, and write the sum below:
3 | -11 | 31 | -21
- 3
0 | -33
Step 4: Repeat the process with the new coefficients. Bring down the -33:
3 | -11 | 31 | -21
- 3
0 | -33 | -54
Step 5: Repeat the process with the new coefficients. Bring down the -54:
3 | -11 | 31 | -21
- 3
0 | -33 | -54 | -75
Step 6: The resulting quotient is the coefficients of the quotient polynomial. In this case, the coefficients are 0, -33, -54, and -75. Therefore, the quotient polynomial is x³ - 33x² - 54x - 75.