A polynomial P(x) with integral coefficients takes the value 5 at four distinct integral values of x. The number of integral solutions of the equation `P(x)-3=0` is ________________

Answer

To solve the problem, we need to analyze the polynomial \( P(x) \) given that it takes the value 5 at four distinct integral values of \( x \). We will denote these four distinct integral values as \( \alpha_1, \alpha_2, \alpha_3, \alpha_4 \).

### Step-by-Step Solution:

1. **Understanding the Polynomial**:
Since \( P(x) \) takes the value 5 at four distinct integral values, we can express this as:
\[ P(\alpha_1) = 5, \quad P(\alpha_2) = 5, \quad P(\alpha_3) = 5, \quad P(\alpha_4) = 5 \]

2. **Constructing a New Polynomial**:
We can construct a new polynomial \( Q(x) = P(x) - 5 \). This polynomial \( Q(x) \) will have roots at \( \alpha_1, \alpha_2, \alpha_3, \alpha_4 \):
\[ Q(x) = (x - \alpha_1)(x - \alpha_2)(x - \alpha_3)(x - \alpha_4) \cdot R(x) \]
where \( R(x) \) is some polynomial with integer coefficients.

3. **Finding the Equation**:
We want to find the integral solutions of the equation \( P(x) - 3 = 0 \). This can be rewritten using \( Q(x) \):
\[ P(x) - 3 = 0 \implies Q(x) + 2 = 0 \implies Q(x) = -2 \]

4. **Analyzing the Roots**:
The polynomial \( Q(x) \) is a polynomial of degree at least 4 (since it has 4 roots). The equation \( Q(x) = -2 \) can have multiple solutions.

5. **Finding the Number of Solutions**:
Since \( Q(x) \) is a polynomial of degree at least 4, it can cross the line \( y = -2 \) at most 4 times, depending on the behavior of the polynomial. Therefore, the number of integral solutions to \( P(x) - 3 = 0 \) can be at most 4.

6. **Conclusion**:
Since \( P(x) \) takes the value 5 at four distinct integral values, and we are looking for the number of times \( P(x) - 3 = 0 \) can occur, the maximum number of integral solutions is 4.

Thus, the number of integral solutions of the equation \( P(x) - 3 = 0 \) is **4**.