Find all the zeros of [tex]$f(x)$[/tex].

[tex]f(x)=x^4 - 3x^3 - 27x^2 - 13x + 42[/tex]

Arrange your answers from smallest to largest. If there is a double root, list it twice.

[tex]x = \ ?[/tex]

1. [tex]$\square$[/tex]
2. [tex]$\square$[/tex]
3. [tex]$\square$[/tex]
4. [tex]$\square$[/tex]



Answer :

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]