To solve the problem using synthetic division and find the quotient and remainder for [tex]\(\left(2x^4 - 3x^3 - 20x - 21\right) \div (x - 3)\)[/tex], follow these steps:
1. Identify the coefficients: The polynomial [tex]\(2x^4 - 3x^3 + 0x^2 - 20x - 21\)[/tex] has the coefficients [2, -3, 0, -20, -21].
2. Set up synthetic division: The divisor is [tex]\(x - 3\)[/tex], which means we use [tex]\(+3\)[/tex] for synthetic division.
3. Perform synthetic division:
- Write down the coefficients: [tex]\([2, -3, 0, -20, -21]\)[/tex]
- Use 3 as the synthetic divisor.
Start the process:
- Bring down the first coefficient (2),
- Multiply it by the synthetic divisor (3) and add this result to the next coefficient.
Here are the steps in detail:
1. First Coefficient: 2
- Bring down [tex]\(2\)[/tex].
2. Next Coefficient Calculation:
- Multiply 2 (current carry down) by 3 (synthetic divisor) = 6
- Add this to the next coefficient (-3): [tex]\(-3 + 6 = 3\)[/tex]
- New coefficient is [tex]\(3\)[/tex].
3. Next Coefficient Calculation:
- Multiply 3 (current coefficient) by 3 (synthetic divisor) = 9
- Add this to the next coefficient (0): [tex]\(0 + 9 = 9\)[/tex]
- New coefficient is [tex]\(9\)[/tex].
4. Next Coefficient Calculation:
- Multiply 9 (current coefficient) by 3 (synthetic divisor) = 27
- Add this to the next coefficient (-20): [tex]\(-20 + 27 = 7\)[/tex]
- New coefficient is [tex]\(7\)[/tex].
5. Next Coefficient Calculation:
- Multiply 7 (current coefficient) by 3 (synthetic divisor) = 21
- Add this to the next coefficient (-21): [tex]\(-21 + 21 = 0\)[/tex]
- New coefficient is [tex]\(0\)[/tex].
4. Determine the Quotient and Remainder:
The results of synthetic division give us the coefficients of the quotient and a remainder, which are:
- Quotient coefficients: [tex]\([2, 3, 9, 7]\)[/tex]
- Remainder: [tex]\(0\)[/tex]
Thus, the quotient polynomial is:
[tex]\[2x^3 + 3x^2 + 9x + 7\][/tex]
So, the correct answer is:
A. [tex]\(2x^3 + 3x^2 + 9x + 7\)[/tex]
