Let's perform synthetic division to test whether [tex]\( x = -1 \)[/tex] is a zero of the polynomial [tex]\( n(x) = 3x^3 - x^2 - 39x + 13 \)[/tex].
### Step-by-Step Synthetic Division for [tex]\( x = -1 \)[/tex]
1. Set up the synthetic division:
Coefficients of the polynomial are: [tex]\( 3, -1, -39, 13 \)[/tex].
2. Use synthetic division format:
```
-1 | 3 -1 -39 13
| -3 4 43
-----------------
3 -4 -35 56
```
3. Perform the synthetic division steps:
- Bring down the first coefficient:
- [tex]\( 3 \)[/tex]
- Multiply [tex]\( 3 \)[/tex] by [tex]\( -1 \)[/tex] and write the result below the next coefficient:
- [tex]\( 3 \times -1 = -3 \)[/tex]
- Add the result to the second coefficient:
- [tex]\( -1 + (-3) = -4 \)[/tex]
- Multiply [tex]\( -4 \)[/tex] by [tex]\( -1 \)[/tex] and write the result below the next coefficient:
- [tex]\( -4 \times -1 = 4 \)[/tex]
- Add the result to the third coefficient:
- [tex]\( -39 + 4 = -35 \)[/tex]
- Multiply [tex]\( -35 \)[/tex] by [tex]\( -1 \)[/tex] and write the result below the next coefficient:
- [tex]\( -35 \times -1 = 35 \)[/tex]
- Add the result to the last coefficient:
- [tex]\( 13 + 35 = 48 \)[/tex]
4. Interpret the final row:
The final row of numbers is: [tex]\( 3, -4, -35, 48 \)[/tex]
The remainder is [tex]\( 48 \)[/tex].
Therefore, [tex]\(-1\)[/tex] is not a zero of [tex]\( n(x) \)[/tex].
