Question

Complex Numbers Unit Test
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Question
One of the solutions to the equation x3+5x2+10x+50=0
is −5
. Find the full solution set. Which of the following options correctly describes the solution set?

Option #1: It has one real solution and two non-real solutions.

Option #2: It has one non-real solution and two real solutions.

Option #3: It has three real solutions.

(1 point)
Option #

Answer 1

To find the full solution set for the polynomial equation \(x^3 + 5x^2 + 10x + 50 = 0\) given that one of the solutions is \(x = -5\), we can perform synthetic division to factor the polynomial.

1. **Perform synthetic division with \(x + 5\):**

Set \(x = -5\) for synthetic division:

```
-5 | 1 5 10 50
| -5 0 -50
---------------------
1 0 10 0
```

The result is \(x^2 + 10\) with a remainder of 0.

2. **Factor the equation:**

Now, we have:
\[
x^3 + 5x^2 + 10x + 50 = (x + 5)(x^2 + 10)
\]

3. **Solve \(x^2 + 10 = 0\):**

\[
x^2 = -10
\]

The solutions are:
\[
x = \sqrt{-10} = \pm i\sqrt{10}
\]

4. **Full solution set:**

The solution set includes:
- One real solution: \(x = -5\)
- Two non-real solutions: \(x = i\sqrt{10}\) and \(x = -i\sqrt{10}\)

5. **Choose the correct option:**

The correct option that describes this solution set is:

**Option #1: It has one real solution and two non-real solutions.**