Question

16 of 2016 of 20 Items



Question
One of the solutions to the equation x3+5x2+10x+50=0
is x=−5
. Find the full solution set. Which of the following options correctly describes the solution set?(1 point)
Responses

It has three non-real solutions
It has three non-real solutions

It has two real solutions and one non-real solution
It has two real solutions and one non-real solution

It has three real solutions
It has three real solutions

It has one real solution and two non-real solutions
It has one real solution and two non-real solutions

Answer 1

To solve the polynomial \( x^3 + 5x^2 + 10x + 50 = 0 \) given that \( x = -5 \) is a solution, we can use synthetic division to factor the polynomial.

1. **Synthetic division of \( x^3 + 5x^2 + 10x + 50 \) by \( x + 5 \):**

- Coefficients: \(1, 5, 10, 50\)
- Using \( -5 \):

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

The result from the synthetic division is \( x^2 + 0x + 10 \) (which simplifies to \( x^2 + 10 \)), and the remainder is 0. Hence, we can write:

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

2. **Finding the roots of \( x^2 + 10 = 0 \):**

To find the roots of \( x^2 + 10 = 0 \), we set it equal to zero:

\[
x^2 = -10
\]

Taking the square root of both sides gives:

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

These are non-real solutions.

3. **Solution set:**

The solutions to the original equation \( x^3 + 5x^2 + 10x + 50 = 0 \) are:

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

Thus, the correct description of the solution set is:

**It has one real solution and two non-real solutions.**