To find all the zeros of the polynomial function [tex]\( f(x) = x^4 - 2x^3 - 7x^2 + 8x + 12 \)[/tex], we need to solve the equation [tex]\( f(x) = 0 \)[/tex]. Here are the steps:
1. Writing the polynomial equation:
[tex]\[
x^4 - 2x^3 - 7x^2 + 8x + 12 = 0
\][/tex]
2. Finding the roots of the polynomial:
Solving the polynomial equation [tex]\( x^4 - 2x^3 - 7x^2 + 8x + 12 = 0 \)[/tex] will give us the values of [tex]\( x \)[/tex] for which the equation holds true (also known as the zeros of the polynomial).
3. Identify the roots:
After solving the polynomial equation, we find that the roots are [tex]\( x = -2 \)[/tex], [tex]\( x = -1 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = 3 \)[/tex].
4. Arranging the roots from smallest to largest:
To list the roots from smallest to largest, we sort the values: [tex]\( -2 \)[/tex], [tex]\( -1 \)[/tex], [tex]\( 2 \)[/tex], and [tex]\( 3 \)[/tex].
So, the zeros of the polynomial [tex]\( f(x) = x^4 - 2x^3 - 7x^2 + 8x + 12 \)[/tex] sorted from smallest to largest are:
[tex]\[ x = -2, -1, 2, 3 \][/tex]
Therefore, listing the roots with the given format:
[tex]\[ x = [-2], -1, 2, 3 \][/tex]
These are the zeros of the polynomial [tex]\( f(x) \)[/tex].
