In the xy-plane, the graph of the polynomial function f crosses the x-axis at exactly two points, (a, 0) and (b, 0), where a and b are both positive. Which of the following could define f ?

Options

A. f (x) = (x − a)(x − b)
B. f (x) = (x + a)(x + b)
C. f (x) = (x − a)(x + b)
D. f (x) = x(x − a)(x − b)

Explanation

To determine which polynomial function \( f \) crosses the x-axis at exactly two points, \( (a, 0) \) and \( (b, 0) \), where both \( a \) and \( b \) are positive, we need to analyze the given options.

### Step-by-step Solution:

1. **Understanding the Problem**:
Since the polynomial crosses the x-axis at exactly two points, this means that \( f(a) = 0 \) and \( f(b) = 0 \). The polynomial must have roots at \( a \) and \( b \) and should not cross the x-axis at any other point.

2. **Form of the Polynomial**:
A polynomial that crosses the x-axis at two distinct points can be represented as:
\[ f(x) = k(x - a)(x - b) \]
where \( k \) is a non-zero constant. This form ensures that the polynomial has roots at \( a \) and \( b \).

3. **Analyzing the Options**:
We will evaluate each option to see if substituting \( a \) and \( b \) results in \( f(a) = 0 \) and \( f(b) = 0 \).

- **Option A**: Check if \( f(a) = 0 \) and \( f(b) = 0 \).
- **Option B**: Substitute \( a \) and \( b \) into the polynomial and check if it equals 0.
- **Option C**: Similarly, substitute \( a \) and \( b \) and check.
- **Option D**: Perform the same substitution.

4. **Substituting Values**:
- For each option, substitute \( a \) and \( b \) into the polynomial and check if the result is zero.
- If any option yields \( f(a) = 0 \) and \( f(b) = 0 \) without any additional roots, it is a candidate.

5. **Identifying the Correct Option**:
- After evaluating all options, we find that:
- **Option A** satisfies \( f(a) = 0 \) and \( f(b) = 0 \) without introducing any other roots.
- Other options either introduce additional roots or do not satisfy the conditions.

### Conclusion:
The polynomial function \( f \) that crosses the x-axis at exactly two points \( (a, 0) \) and \( (b, 0) \) is defined by **Option A**.