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Answer :
Let's find the remainder when dividing the polynomial by using synthetic division.
### Problem:
Divide the polynomial [tex]\( p(x) = x^3 + 2x^2 - 3x + 2 \)[/tex] by [tex]\( x - 1 \)[/tex] and find the remainder.
### Solution:
1. Set up the synthetic division:
- The root for division is 1 (because we are dividing by [tex]\( x - 1 \)[/tex]).
- The coefficients of the polynomial are: 1, 2, -3, and 2.
2. Perform synthetic division:
Write the root (1 in this case) on the left side, and the coefficients of the polynomial in a row:
```
1 | 1 2 -3 2
|
```
3. Bring down the leading coefficient:
- The leading coefficient is 1, so bring it down below the line:
```
1 | 1 2 -3 2
|
-----------------
1
```
4. Multiply and add:
- Multiply the number below the line (1) by the root (1) and write the result under the next coefficient (2). Then, add:
```
1 | 1 2 -3 2
| 1
-----------------
1 3
```
- Multiply the number just written (3) by the root (1) and write the result under the next coefficient (-3). Then, add:
```
1 | 1 2 -3 2
| 1 3
-----------------
1 3 0
```
- Multiply the number just written (0) by the root (1) and write the result under the next coefficient (2). Then, add:
```
1 | 1 2 -3 2
| 1 3 0
-----------------
1 3 0 2
```
5. Identify the remainder:
- The last number, 2, is the remainder of the division.
Therefore, the remainder when dividing the polynomial [tex]\( x^3 + 2x^2 - 3x + 2 \)[/tex] by [tex]\( x - 1 \)[/tex] is [tex]\(\boxed{2}\)[/tex].
### Problem:
Divide the polynomial [tex]\( p(x) = x^3 + 2x^2 - 3x + 2 \)[/tex] by [tex]\( x - 1 \)[/tex] and find the remainder.
### Solution:
1. Set up the synthetic division:
- The root for division is 1 (because we are dividing by [tex]\( x - 1 \)[/tex]).
- The coefficients of the polynomial are: 1, 2, -3, and 2.
2. Perform synthetic division:
Write the root (1 in this case) on the left side, and the coefficients of the polynomial in a row:
```
1 | 1 2 -3 2
|
```
3. Bring down the leading coefficient:
- The leading coefficient is 1, so bring it down below the line:
```
1 | 1 2 -3 2
|
-----------------
1
```
4. Multiply and add:
- Multiply the number below the line (1) by the root (1) and write the result under the next coefficient (2). Then, add:
```
1 | 1 2 -3 2
| 1
-----------------
1 3
```
- Multiply the number just written (3) by the root (1) and write the result under the next coefficient (-3). Then, add:
```
1 | 1 2 -3 2
| 1 3
-----------------
1 3 0
```
- Multiply the number just written (0) by the root (1) and write the result under the next coefficient (2). Then, add:
```
1 | 1 2 -3 2
| 1 3 0
-----------------
1 3 0 2
```
5. Identify the remainder:
- The last number, 2, is the remainder of the division.
Therefore, the remainder when dividing the polynomial [tex]\( x^3 + 2x^2 - 3x + 2 \)[/tex] by [tex]\( x - 1 \)[/tex] is [tex]\(\boxed{2}\)[/tex].
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