To find the zeros of the polynomial function [tex]\( f(x) = x^4 - 7x^3 - 39x^2 - 53x - 22 \)[/tex], we need to solve the equation [tex]\( f(x) = 0 \)[/tex]. The roots of this equation represent the values of [tex]\( x \)[/tex] for which the polynomial is equal to zero.
1. Formulate the Polynomial Equation:
[tex]\[
f(x) = x^4 - 7x^3 - 39x^2 - 53x - 22
\][/tex]
2. Solve the Polynomial Equation:
Solving the polynomial equation [tex]\( x^4 - 7x^3 - 39x^2 - 53x - 22 = 0 \)[/tex] involves finding the values of [tex]\( x \)[/tex] (roots) where the equation holds true.
3. Identify the Roots:
The roots of the polynomial equation are the values where the function [tex]\( f(x) \)[/tex] is zero. These values can be obtained (for example, by factoring, using numerical methods, or leveraging symbolic computation).
The roots of the polynomial [tex]\( x^4 - 7x^3 - 39x^2 - 53x - 22 \)[/tex] are:
[tex]\[
x = -2, -1, 11
\][/tex]
4. Arrange the Roots in Ascending Order:
To write the roots from smallest to largest, we list them in increasing order:
[tex]\[
x = -2, -1, 11
\][/tex]
Therefore, the zeros of the polynomial function [tex]\( f(x) = x^4 - 7x^3 - 39x^2 - 53x - 22 \)[/tex] are:
[tex]\[
x = [-2, -1, 11]
\][/tex]
