Question

complex numbers & higher order polynomials practice
complete this assessment to review what you’ve learned. it will not count toward your grade.
the equation ( x^4 + x^3 - 3x^2 + 9x - 108 = 0 ) can be factored as ( (x^2 + x - 12)(x^2 + 9) = 0 ). find the full solution set. which of the following options correctly describes the solution set?
option #1 it has one real solution and three non - real solutions.
option #2 it has two real solutions and two non - real solutions.
option #3 it has four real solutions.
option #4 it has four non - real solutions.
(1 point)

Explanation:

Step1: Solve \(x^2 + x - 12 = 0\)

Factor the quadratic: \(x^2 + x - 12=(x + 4)(x - 3)=0\).
Set each factor to zero: \(x + 4 = 0\) gives \(x=-4\); \(x - 3 = 0\) gives \(x = 3\). These are real solutions.

Step2: Solve \(x^2 + 9 = 0\)

Rearrange: \(x^2=-9\). Take square roots: \(x=\pm\sqrt{-9}=\pm 3i\). These are non - real (complex) solutions.

Step3: Count solutions

From \(x^2 + x - 12 = 0\), we have 2 real solutions. From \(x^2 + 9 = 0\), we have 2 non - real solutions. So total is 2 real and 2 non - real solutions.

Answer:

Option #2: It has two real solutions and two non - real solutions.