To solve the polynomial division [tex]\((2 x^4 - 3 x^3 - 20 x - 21) \div (x - 3)\)[/tex] using synthetic division, follow these steps:
1. Identify the coefficients of the polynomial: The polynomial is [tex]\(2 x^4 - 3 x^3 + 0 x^2 - 20 x - 21\)[/tex]. The coefficients are [tex]\([2, -3, 0, -20, -21]\)[/tex].
2. Identify the root of the divisor: The divisor is [tex]\(x - 3\)[/tex], so the root is [tex]\(3\)[/tex].
3. Set up the synthetic division table:
```
| 3 |
-------------
2 | |
-3 | |
0 | |
-20 | |
-21 | |
```
4. Perform the synthetic division steps:
- Bring down the first coefficient (2) as it is.
- Multiply the root (3) by this value, and add it to the next coefficient (-3):
- [tex]\[ 3 \times 2 = 6 \][/tex]
- [tex]\[ -3 + 6 = 3 \][/tex]
- Repeat the same for the next coefficients:
- [tex]\[ 3 \times 3 = 9 \][/tex]
- [tex]\[ 0 + 9 = 9 \][/tex]
- [tex]\[ 3 \times 9 = 27 \][/tex]
- [tex]\[ -20 + 27 = 7 \][/tex]
- [tex]\[ 3 \times 7 = 21 \][/tex]
- [tex]\[ -21 + 21 = 0 \][/tex]
5. Form the quotient and the remainder:
- The quotient coefficients are the result from the synthetic division, except for the remainder, which is the last value obtained.
- The quotient polynomial is [tex]\(2 x^3 + 3 x^2 + 9 x + 7\)[/tex].
- The remainder is [tex]\(0\)[/tex].
Based on these steps, the result for the synthetic division is:
[tex]\[
\boxed{2 x^3 + 3 x^2 + 9 x + 7}
\][/tex]
Matching this with the options provided, the correct answer is:
A. [tex]\(2 x^3 + 3 x^2 + 9 x + 7\)[/tex].
