If a is a zero of f(x) then, _____ is one of the factors of f(x).

Options

A. (x-2a)
B. (x-a)
C. (x+a)
D. (2x-a)

Answer

To solve the question, we need to determine which expression is a factor of the polynomial \( f(x) \) given that \( a \) is a zero of \( f(x) \).

### Step-by-Step Solution:

1. **Understanding the Concept of Zeroes and Factors**:
- If \( a \) is a zero of \( f(x) \), it means that when we substitute \( x = a \) into the polynomial \( f(x) \), the result is zero. This can be mathematically expressed as:
\[ f(a) = 0 \]

2. **Identifying the Factor**:
- According to the Factor Theorem, if \( a \) is a zero of \( f(x) \), then \( (x - a) \) is a factor of \( f(x) \). This means that \( f(x) \) can be expressed in the form:
\[ f(x) = (x - a) \cdot g(x) \]
where \( g(x) \) is another polynomial.

3. **Verifying with Options**:
- We need to check the provided options to see which one satisfies the condition \( f(a) = 0 \).
- Let's analyze the options:
- **Option 1**: \( x - 2a \)
- Substitute \( a \):
\[ f(a) = a - 2a = -a \quad (\text{not zero}) \]
- **Option 2**: \( x - a \)
- Substitute \( a \):
\[ f(a) = a - a = 0 \quad (\text{is zero}) \]
- **Option 3**: \( x + a \)
- Substitute \( a \):
\[ f(a) = a + a = 2a \quad (\text{not zero}) \]
- **Option 4**: \( 2x - a \)
- Substitute \( a \):
\[ f(a) = 2a - a = a \quad (\text{not zero}) \]

4. **Conclusion**:
- From the analysis, only **Option 2** \( (x - a) \) satisfies the condition \( f(a) = 0 \). Therefore, if \( a \) is a zero of \( f(x) \), then \( (x - a) \) is one of the factors of \( f(x) \).

### Final Answer:
If \( a \) is a zero of \( f(x) \), then \( (x - a) \) is one of the factors of \( f(x) \).