Let's start by finding the factors of the function [tex]\( f(x) = 2x^4 - x^3 - 18x^2 + 9x \)[/tex].
Step 1: Factor the polynomial
The function [tex]\( f(x) \)[/tex] can be factored into the product of its linear factors. The factored form of the polynomial is:
[tex]\[
f(x) = x (x - 3) (x + 3) (2x - 1)
\][/tex]
Step 2: Identify the zeros from the factored form
To find the zeros of the function, set each factor equal to zero and solve for [tex]\( x \)[/tex]:
1. [tex]\( x = 0 \)[/tex]
2. [tex]\( x - 3 = 0 \)[/tex]
[tex]\[
x = 3
\][/tex]
3. [tex]\( x + 3 = 0 \)[/tex]
[tex]\[
x = -3
\][/tex]
4. [tex]\( 2x - 1 = 0 \)[/tex]
[tex]\[
2x = 1 \implies x = \frac{1}{2}
\][/tex]
Step 3: List the zeros in ascending order
The zeros of [tex]\( f(x) \)[/tex] from left to right are:
- [tex]\( x = -3 \)[/tex]
- [tex]\( x = 0 \)[/tex]
- [tex]\( x = \frac{1}{2} \)[/tex]
- [tex]\( x = 3 \)[/tex]
Therefore, the completed statement is:
From left to right, function [tex]\( f \)[/tex] has zeros at [tex]\( x = -3 \)[/tex], [tex]\( x = 0 \)[/tex], [tex]\( x = \frac{1}{2} \)[/tex], and [tex]\( x = 3 \)[/tex].
So the boxed answers should be filled in as follows:
[tex]\[
x = -3, x = 0, x = \frac{1}{2}, x = 3
\][/tex]
