To solve the polynomial equation [tex]\(x^3 - x^2 - 36x + 36 = 0\)[/tex] and determine the number of possible imaginary (complex) solutions, follow these steps:
1. Set up the equation: We start by examining the polynomial given:
[tex]\[
x^3 - x^2 - 36x + 36 = 0
\][/tex]
2. Find the roots of the polynomial: Through solving this cubic equation, we obtain the roots of the equation. For this specific polynomial, the roots are found to be:
[tex]\[
x = -6, \quad x = 1, \quad x = 6
\][/tex]
3. Classify the roots: Next, we need to determine the nature of these roots. A complex (imaginary) root in this context means a root that has a non-zero imaginary component. When examining the roots provided:
[tex]\[
-6, \quad 1, \quad 6
\][/tex]
we notice that all these numbers are real numbers.
4. Determine the number of imaginary solutions: Since none of the roots have an imaginary component, we conclude that the number of imaginary (complex) solutions is:
[tex]\[
0
\][/tex]
Thus, the polynomial equation [tex]\(x^3 - x^2 - 36x + 36 = 0\)[/tex] has 0 imaginary (complex) solutions. The correct answer is:
```
(0)
```
