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We begin by first looking for rational zeros. We can apply the rational zero theorem because the polynomial has integer coefficients.
[tex]\[
n(x) = 3x^3 - x^2 - 39x + 13
\][/tex]

Possible rational zeros:
Factors of 13
[tex]\[
= \pm 1, \pm \frac{1}{3}, \pm 13, \pm \frac{13}{3}
\][/tex]

Part 2 of 6

Next, use synthetic division and the remainder theorem to determine if any of the numbers in the list is a zero of [tex]\(n\)[/tex].

Test [tex]\( x = 1 \)[/tex]:
The remainder is -24.
Therefore, 1 is not a zero of [tex]\( n(x) \)[/tex].

Part 3 of 6

Test [tex]\( x = -1 \)[/tex]:
The remainder is [tex]\(\square\)[/tex].
Therefore, -1 (Choose one) [tex]\( \square \)[/tex] a zero of [tex]\( n(x) \)[/tex].



Answer :

GinnyAnswer
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].

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