The number of polynomials p(x) with integer coefficients such that the curve y = p(x) passes through (2, 2) and (4, 5) is

Options

A. 0
B. 1
C. more than 1 but finite
D. infinite

Explanation

To find the number of polynomials \( p(x) \) with integer coefficients such that the curve \( y = p(x) \) passes through the points \( (2, 2) \) and \( (4, 5) \), we can follow these steps:

### Step 1: Set up the equations
We know that \( p(2) = 2 \) and \( p(4) = 5 \). We can express this as two equations:
1. \( p(2) = a_0 + a_1 \cdot 2 + a_2 \cdot 2^2 + \ldots + a_n \cdot 2^n = 2 \)
2. \( p(4) = a_0 + a_1 \cdot 4 + a_2 \cdot 4^2 + \ldots + a_n \cdot 4^n = 5 \)

### Step 2: Subtract the equations
We can subtract the first equation from the second:
\[ p(4) - p(2) = 5 - 2 = 3 \]
This gives us:
\[ (a_0 + a_1 \cdot 4 + a_2 \cdot 4^2 + \ldots + a_n \cdot 4^n) - (a_0 + a_1 \cdot 2 + a_2 \cdot 2^2 + \ldots + a_n \cdot 2^n) = 3 \]

### Step 3: Simplify the equation
This simplifies to:
\[ a_1(4 - 2) + a_2(4^2 - 2^2) + a_3(4^3 - 2^3) + \ldots + a_n(4^n - 2^n) = 3 \]
This can be further simplified:
\[ 2a_1 + (16 - 4)a_2 + (64 - 8)a_3 + \ldots + (4^n - 2^n)a_n = 3 \]
or
\[ 2a_1 + 12a_2 + 56a_3 + \ldots + (4^n - 2^n)a_n = 3 \]

### Step 4: Analyze the coefficients
The left-hand side of the equation is a linear combination of the coefficients \( a_1, a_2, a_3, \ldots, a_n \) with integer coefficients. The right-hand side is 3.

### Step 5: Determine the number of integer solutions
The equation \( 2a_1 + 12a_2 + 56a_3 + \ldots + (4^n - 2^n)a_n = 3 \) can have multiple integer solutions depending on the values of \( a_1, a_2, a_3, \ldots, a_n \). However, since \( 2a_1 \) is even, the left-hand side must also be even, which implies that \( 3 \) cannot be expressed as a sum of even integers.

Thus, there are no integer coefficients \( a_1, a_2, a_3, \ldots, a_n \) that satisfy this equation.

### Conclusion
The number of polynomials \( p(x) \) with integer coefficients that pass through the points \( (2, 2) \) and \( (4, 5) \) is **0**.