To evaluate [tex]\( P(x) = x^3 - x^2 + x + 5 \)[/tex] at [tex]\( c = 1 \)[/tex] using synthetic division and the Remainder Theorem, follow these steps:
1. Write down the coefficients of the polynomial: For the polynomial [tex]\( P(x) = x^3 - x^2 + x + 5 \)[/tex], the coefficients are [tex]\( [1, -1, 1, 5] \)[/tex].
2. Set up the synthetic division: The value [tex]\( c = 1 \)[/tex] will be used in the synthetic division process. Write [tex]\( c \)[/tex] to the left and the coefficients to the right.
```
1 | 1 -1 1 5
```
3. Bring down the leading coefficient: The first coefficient (which is 1) is brought down unchanged:
```
1 | 1 -1 1 5
----
| 1
```
4. Perform the synthetic division steps:
- Multiply the value of [tex]\( c \)[/tex] (which is 1) by the number just written below the line (initially 1), and write the result under the next coefficient:
```
1 | 1 -1 1 5
| 1
----
| 1 1
```
- Add the number above to the number just written below the line:
```
1 | 1 -1 1 5
| 1
----
| 1 0
```
- Repeat these steps for the next coefficients:
- Multiply 1 (c) by 0:
```
1 | 1 -1 1 5
| 1 0
----
| 1 0 1
```
- Add -1 + 1:
```
1 | 1 -1 1 5
| 1 0
----
| 1 0 1
```
- Multiply 1 by 1:
```
1 | 1 -1 1 5
| 1 0 1
----
| 1 0 1 6
```
5. Identify the remainder: The last number written is the remainder.
In this case, the remainder is 6.
According to the Remainder Theorem, the remainder when [tex]\( P(x) \)[/tex] is divided by [tex]\( x - c \)[/tex] is [tex]\( P(c) \)[/tex].
So, [tex]\( P(1) = 6 \)[/tex].
