Certainly! Let's break down the problem step by step to identify the zeros and eventually the factors of the function [tex]\( f(x) \)[/tex].
Given:
[tex]\[ f(x) = 2x^4 - x^3 - 18x^2 + 9x \][/tex]
### Step 1: Identifying the Zeros
The zeros of the function are:
[tex]\[ x = -3, \quad x = 0, \quad x = \frac{1}{2}, \quad \text{and} \quad x = 3 \][/tex]
### Step 2: Writing the Factors
Each zero corresponds to a factor of the polynomial. Specifically:
- For [tex]\( x = -3 \)[/tex], the factor is [tex]\( (x + 3) \)[/tex]
- For [tex]\( x = 0 \)[/tex], the factor is [tex]\( x \)[/tex]
- For [tex]\( x = \frac{1}{2} \)[/tex], the factor is [tex]\( \left(x - \frac{1}{2}\right) \)[/tex]. However, to keep the polynomial with integer coefficients, we rewrite this factor as [tex]\( \left(2x - 1\right) \)[/tex]
- For [tex]\( x = 3 \)[/tex], the factor is [tex]\( (x - 3) \)[/tex]
### Step 3: Completing the Statement
With these zeros, we can list them from left to right and include them in the statement.
From left to right, function [tex]\( f \)[/tex] has zeros at:
[tex]\[ x = -3, \quad x = 0, \quad x = \frac{1}{2}, \quad \text{and} \quad x = 3 \][/tex]
Now fill in the boxes with the corresponding zeros:
[tex]\[ f \text{ has zeros at } x = -3, \quad x = 0, \quad x = \frac{1}{2}, \quad \text{and} \quad x = 3 \][/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].
