Question

One of the solutions to the equation x3+5x2+10x+50=0
𝑥
3
+
5
𝑥
2
+
10
𝑥
+
50
=
0
is −5
−
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.

Answer 1

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

### Step 1: Synthetic division

Perform synthetic division with \( x = -5 \):

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

This gives us the quotient \( x^2 + 10 \) (the remainder is \( 0 \)), meaning we can factor the original cubic equation as:

\[
x^3 + 5x^2 + 10x + 50 = (x + 5)(x^2 + 10)
\]

### Step 2: Solve the quadratic factor

Now, we solve the quadratic equation \( x^2 + 10 = 0 \):

\[
x^2 = -10
\]

Taking the square root of both sides, we get:

\[
x = \pm i\sqrt{10}
\]

### Step 3: Summary of solutions

Thus, the full solution set for the equation \( x^3 + 5x^2 + 10x + 50 = 0 \) is:

\[
x = -5, \quad x = i\sqrt{10}, \quad x = -i\sqrt{10}
\]

### Step 4: Identify the nature of the solutions

- One real solution: \( -5 \)
- Two non-real solutions: \( i\sqrt{10} \) and \( -i\sqrt{10} \)

### Conclusion:

The correct option that describes the solution set is:

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