To determine the nature of the zeros of the polynomial function [tex]\( f(x) = x^3 + x^2 - 8x - 8 \)[/tex], we need to find the roots and identify which are real and which are nonreal.
1. Formulate the Polynomial: We start with the given polynomial:
[tex]\[
f(x) = x^3 + x^2 - 8x - 8
\][/tex]
2. Find the Roots: By solving this polynomial equation for [tex]\( x \)[/tex], we determine the roots.
3. Classify the Roots: We need to classify these roots into real and nonreal (complex) zeros:
- A root is real if it has no imaginary part.
- A root is nonreal if it has an imaginary part.
4. Count the Real and Nonreal Zeros: Upon solving the polynomial, we find the roots and categorize them accordingly.
For this particular polynomial function, the roots have been found and categorized as follows:
- All three zeros of the polynomial [tex]\( x^3 + x^2 - 8x - 8 \)[/tex] are real.
- There are no nonreal (complex) zeros.
Thus, the correct answer is:
- The function has three real zeros. The graph of the function intersects the [tex]\( x \)[/tex]-axis at three locations.
