To find the zeros of the polynomial function
[tex]\[ f(x) = x^4 - 3x^3 - 27x^2 - 13x + 42, \][/tex]
we need to solve for [tex]\( x \)[/tex] such that [tex]\( f(x) = 0 \)[/tex].
Here is a detailed, step-by-step solution:
1. Write the polynomial: Start with the given polynomial:
[tex]\[ f(x) = x^4 - 3x^3 - 27x^2 - 13x + 42 \][/tex]
2. Find the roots (zeros) of the polynomial: This involves solving the equation [tex]\( f(x) = 0 \)[/tex].
3. List the solutions: The solutions to the polynomial equation are the roots or zeros of [tex]\( f(x) \)[/tex].
4. Arrange the zeros in ascending order: Once the roots are found, they should be arranged from smallest to largest.
From the calculations, we find the zeros of the polynomial [tex]\( f(x) = 0 \)[/tex] to be:
[tex]\[ x = -3, -2, 1, 7 \][/tex]
5. Arrange these zeros from smallest to largest:
[tex]\[ x = [-3, -2, 1, 7] \][/tex]
So, the zeros of the polynomial function [tex]\( f(x) \)[/tex] arranged from smallest to largest are:
[tex]\[ x = -3, -2, 1, 7 \][/tex]
Thus, the final answer is:
[tex]\[
\begin{aligned}
& \boxed{-3} \\
& \boxed{-2} \\
& \boxed{1} \\
& \boxed{7}
\end{aligned}
\][/tex]
