one of the solutions to the equations x^3+5x^2+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.
Option #1: It has one real solution and two non-real solutions.
The given equation is a cubic equation, and a cubic equation can have at most three solutions. We know that one of the solutions is x = -5.
To find the other solutions, we can use synthetic division or polynomial long division to divide the given equation by (x + 5). Dividing x^3 + 5x^2 + 10x + 50 by (x + 5) gives us x^2 + 5x + 10.
Now, we can solve the quadratic equation x^2 + 5x + 10 = 0. We can use the quadratic formula or complete the square to find its solutions.
Using the quadratic formula, we have x = (-5 ± √(-15))/2. Since -15 is negative, the square root of -15 will be a non-real complex number.
Therefore, we have one real solution (x = -5) and two non-real solutions. Thus, the correct option is option #1: It has one real solution and two non-real solutions.