To find the quotient and remainder of the division [tex]\(\frac{2x^2 - 3x + 1}{x - 2}\)[/tex] using synthetic division, follow these steps:
1. Identify the coefficients of the numerator polynomial: The polynomial [tex]\(2x^2 - 3x + 1\)[/tex] has coefficients 2, -3, and 1.
2. Identify the root of the divisor: The divisor [tex]\(x - 2\)[/tex] has a root of [tex]\(2\)[/tex].
3. Set up the synthetic division:
- Write down the coefficients of the polynomial: 2, -3, 1.
- Write the root of the divisor (2) to the left.
4. Perform synthetic division:
- Bring down the first coefficient (2) as is.
- Multiply this 2 by the root (2) and write the result (4) under the next coefficient (-3).
- Add -3 and 4 to get 1, and write 1 below the line.
- Multiply this 1 by the root (2) and write the result (2) under the next coefficient (1).
- Add 1 and 2 to get 3, and write 3 below the line. This is the remainder.
The synthetic division setup and steps look like this:
```
2 | 2 -3 1
___________
| 2 1 3
```
- The quotient coefficients are the numbers before the last value: [2, 1].
- The last number is the remainder: 3.
5. Write the final result:
- The quotient is [tex]\([2, 1]\)[/tex], which corresponds to [tex]\(2x + 1\)[/tex].
- The remainder is [tex]\(3\)[/tex].
So, the quotient is [tex]\(2x + 1\)[/tex], and the remainder is [tex]\(3\)[/tex].
Therefore, the quotient is [tex]\(\boxed{2x + 1}\)[/tex] and the remainder is [tex]\(\boxed{3}\)[/tex].
