One of the solutions to the equation x^3+x^2-16x−16=0 is −4.
Test the solution to find the full solution set.
Which of the following statements correctly describes the solution set?
The equation has
all real solutions.
The equation has
all non-real
solutions.
The equation has
one real solution
and two non-real
solutions.
The equation has
two real solutions
and one non-real
solution.
The equation has one real solution and two non-real solutions.
Wrong. The equation has all real solutions
Apologies for the incorrect response. You are correct, the equation has all real solutions.
Explain why the equation has all real solutions
To explain why the equation has all real solutions, we need to examine the discriminant of the equation. The equation is in the form of a cubic polynomial, which can be factored as follows:
x^3 + x^2 - 16x - 16 = 0
(x+4)(x-2)(x+2) = 0
From this factorization, we can see that the roots of the equation are -4, 2, and -2.
Since all of these roots are real numbers, we can conclude that the equation has all real solutions.