Answer :

To determine if [tex]\(x + 3\)[/tex] is a factor of [tex]\(7x^3 + 27x^2 + 9x - 27\)[/tex], we can use synthetic division or polynomial division. Here, I'll use synthetic division to keep it straightforward.

1. Identify the root: Since [tex]\(x + 3\)[/tex] is a factor, we solve [tex]\(x + 3 = 0\)[/tex] which gives [tex]\(x = -3\)[/tex]. This root will be used in synthetic division.

2. Set up the synthetic division: Write down the coefficients of the polynomial [tex]\(7x^3 + 27x^2 + 9x - 27\)[/tex]:
[tex]\[
7, \quad 27, \quad 9, \quad -27
\][/tex]

3. Perform the synthetic division:
- Write [tex]\(-3\)[/tex] on the left and the coefficients on the top row:
```
-3 | 7 27 9 -27
|______________
```

- Bring down the first coefficient (7):
```
-3 | 7 27 9 -27
| 21
|_____________
7 6
```

- Multiply [tex]\(-3\)[/tex] by 7 and place the result under the next coefficient (27):
```
-3 | 7 27 9 -27
| -21
|_____________
7 6
```

- Add the numbers in the second column (27 + (-21)):
```
-3 | 7 27 9 -27
| -21
|_____________
7 6 -9
```

- Continue this process:
```
-3 | 7 27 9 -27
| -21 -15 54
|_________________
7 6 -9 27
```

4. Check the remainder: The last number in the bottom row is the remainder. If the remainder is 0, then [tex]\(x + 3\)[/tex] is a factor of [tex]\(7x^3 + 27x^2 + 9x - 27\)[/tex]. In our case, the remainder is 0.

Since the synthetic division resulted in a remainder of 0, we can conclude that:
[tex]\[
x + 3\text{ is indeed a factor of }7x^3 + 27x^2 + 9x - 27.
\][/tex]

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Rewritten by : Barada