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?%0D%0A%0D%0AOption #1: It has one real solution and two non-real solutions.%0D%0A%0D%0AOption #2: It has one non-real solution and two real solutions. %0D%0A%0D%0AOption #3: It has three real solutions.

Answer 1

To determine the solution set of the polynomial equation \( x^3 + 5x^2 + 10x + 50 = 0 \), knowing that one of the solutions is \( x = -5 \), we can factor the polynomial using synthetic division.

Step 1: Synthetic Division

We will divide the polynomial by \( x + 5 \):

1. Write down the coefficients: \( 1, 5, 10, 50 \).
2. Use \( -5 \) in the synthetic division process.

\[ \begin{array}{r|rrrr} -5 & 1 & 5 & 10 & 50 \ & & -5 & 0 & -50 \ \hline & 1 & 0 & 10 & 0 \ \end{array} \]

Step 2: Result of the Division

The synthetic division shows that:

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

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

Next, set \( x^2 + 10 = 0 \):

\[ x^2 = -10 \]

Taking the square root:

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

Solution Set

The roots of the original polynomial are:

1. \( x = -5 \) (real solution)
2. \( x = i\sqrt{10} \) (non-real solution)
3. \( x = -i\sqrt{10} \) (non-real solution)

Conclusion

The full solution set consists of one real solution and two non-real solutions. Therefore, the correct option is:

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