Answer :
Step-by-step explanation:
To find the zeros and multiplicities of the function \( f(x) = 3x(x-2)(4x-5) \), we follow these steps:
1. **Factorize the function:**
\[ f(x) = 3x(x-2)(4x-5) \]
This function is already factored into linear factors.
2. **Identify the zeros:**
Zeros of the function \( f(x) \) are values of \( x \) for which \( f(x) = 0 \).
- \( x = 0 \)
- \( x - 2 = 0 \Rightarrow x = 2 \)
- \( 4x - 5 = 0 \Rightarrow 4x = 5 \Rightarrow x = \frac{5}{4} \)
So, the zeros of \( f(x) \) are \( x = 0 \), \( x = 2 \), and \( x = \frac{5}{4} \).
3. **Find the multiplicities:**
Multiplicity refers to how many times a zero appears as a factor in the function.
- **Multiplicity of \( x = 0 \):** From \( x \), it appears once as \( x \).
- **Multiplicity of \( x = 2 \):** From \( x-2 \), it appears once as \( x-2 \).
- **Multiplicity of \( x = \frac{5}{4} \):** From \( 4x-5 \), it appears once as \( 4x-5 \).
Therefore, all zeros \( x = 0 \), \( x = 2 \), and \( x = \frac{5}{4} \) have a multiplicity of 1 each.
In summary:
- Zeros: \( x = 0 \), \( x = 2 \), \( x = \frac{5}{4} \)
- Multiplicities: Each zero has a multiplicity of \( \boxed{1} \).