Complex Numbers Unit Test

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.