Let a,b be integers and f(x) be a polynomial with integer coefficients such that f(b)-f(a)=1. Then, the value of b-a, is

Options

A. 1
B. -1
C. 1,-1
D. 0,1

Answer

To solve the problem, we need to determine the value of \( b - a \) given that \( f(b) - f(a) = 1 \) for a polynomial \( f(x) \) with integer coefficients, where \( a \) and \( b \) are integers.

### Step-by-Step Solution:

1. **Assume a Polynomial Form**:
Let's consider \( f(x) \) to be a polynomial of degree 1, which can be expressed as:
\[ f(x) = mx + n \]
where \( m \) and \( n \) are integers.

2. **Calculate \( f(b) - f(a) \)**:
We need to calculate \( f(b) \) and \( f(a) \):
\[ f(b) = mb + n \]
\[ f(a) = ma + n \]
Therefore, the difference is:
\[ f(b) - f(a) = (mb + n) - (ma + n) = mb - ma = m(b - a) \]

3. **Set the Difference Equal to 1**:
According to the problem statement:
\[ f(b) - f(a) = 1 \]
Substituting the expression we found:
\[ m(b - a) = 1 \]

4. **Solve for \( b - a \)**:
Rearranging gives:
\[ b - a = \frac{1}{m} \]
Since \( b - a \) must be an integer (as \( a \) and \( b \) are integers), \( m \) must be a divisor of 1. The integer divisors of 1 are \( 1 \) and \( -1 \).

5. **Determine Possible Values for \( b - a \)**:
- If \( m = 1 \), then:
\[ b - a = \frac{1}{1} = 1 \]
- If \( m = -1 \), then:
\[ b - a = \frac{1}{-1} = -1 \]

6. **Conclusion**:
The only possible integer values for \( b - a \) are \( 1 \) and \( -1 \). Thus, we conclude:
\[ b - a = \pm 1 \]

### Final Answer:
The value of \( b - a \) is \( 1 \) or \( -1 \).