To find the zeros of the polynomial [tex]\( f(x) = x^4 - 7x^3 - 39x^2 - 53x - 22 \)[/tex], we proceed by determining the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex].
The steps to find the roots are as follows:
1. Define the polynomial:
[tex]\[
f(x) = x^4 - 7x^3 - 39x^2 - 53x - 22
\][/tex]
2. Determine the roots: Solving the equation [tex]\( f(x) = 0 \)[/tex], we get the roots of the polynomial, which can have real numbers (including repeated roots).
3. List and arrange the roots: Once the roots are found, they must be arranged from smallest to largest, and if there is a double root, it should be listed twice.
The roots of the polynomial are:
[tex]\[
x = -2, -1, -1, 11
\][/tex]
Arranged from smallest to largest, the zeros of the function [tex]\( f(x) \)[/tex] are:
[tex]\[
x = \{-2, -1, -1, 11\}
\][/tex]
