To find the zeros of the polynomial function
[tex]\[ f(x) = x^3 + 7x^2 - 9x - 63, \][/tex]
we look for the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. These values are known as the roots or zeros of the polynomial. Let's break this down step by step:
1. Understanding the Polynomial:
The polynomial given is a cubic polynomial (degree 3). Therefore, it can have up to three real roots.
2. Setting Up the Equation:
We set the polynomial equal to zero to find its roots:
[tex]\[ x^3 + 7x^2 - 9x - 63 = 0. \][/tex]
3. Finding the Zeros:
We need to solve this cubic equation. The roots of the polynomial are the values of [tex]\( x \)[/tex] that satisfy the equation above.
4. Identifying the Zeros:
Through the process of solving the cubic equation, we determine the values of [tex]\( x \)[/tex] that make the equation true. The polynomial [tex]\( f(x) \)[/tex] is equal to zero at the following points:
[tex]\[ x = -7, \][/tex]
[tex]\[ x = -3, \][/tex]
[tex]\[ x = 3. \][/tex]
Hence, the zeros of the polynomial function [tex]\( f(x) = x^3 + 7x^2 - 9x - 63 \)[/tex] are:
[tex]\[ \boxed{-7, -3, \text{ and } 3}. \][/tex]
