If `A={p:p=((n+2)(2n^(5)+3n^(4)+4n^(3)+5n^(2)+6))/(n^(2)+2n), n p in Z^(+)}` then the number of elements in the set A, is

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the problem, we need to analyze the given set \( A \) defined by the expression for \( p \):

\[ A = \{ p : p = \frac{(n+2)(2n^5 + 3n^4 + 4n^3 + 5n^2 + 6)}{n^2 + 2n}, n \in \mathbb{Z}^+ \} \]

### Step 1: Simplify the Expression for \( p \)

First, we can simplify the expression for \( p \):

\[ p = \frac{(n+2)(2n^5 + 3n^4 + 4n^3 + 5n^2 + 6)}{n^2 + 2n} \]

Notice that we can factor the denominator:

\[ n^2 + 2n = n(n + 2) \]

Now, we can rewrite \( p \):

\[ p = \frac{(n+2)(2n^5 + 3n^4 + 4n^3 + 5n^2 + 6)}{n(n + 2)} \]

We can cancel \( (n + 2) \) from the numerator and denominator (assuming \( n \neq -2 \), which is valid since \( n \) is positive):

\[ p = \frac{2n^5 + 3n^4 + 4n^3 + 5n^2 + 6}{n} \]

### Step 2: Further Simplify \( p \)

Now, we can divide each term in the numerator by \( n \):

\[ p = 2n^4 + 3n^3 + 4n^2 + 5n + \frac{6}{n} \]

### Step 3: Determine Conditions for \( p \) to be a Positive Integer

For \( p \) to be a positive integer, \( \frac{6}{n} \) must also be a positive integer. This means \( n \) must be a divisor of 6.

The positive divisors of 6 are \( 1, 2, 3, \) and \( 6 \).

### Step 4: Calculate \( p \) for Each Valid \( n \)

Now, we will calculate \( p \) for each valid \( n \):

1. For \( n = 1 \):
\[ p = 2(1^4) + 3(1^3) + 4(1^2) + 5(1) + 6 = 2 + 3 + 4 + 5 + 6 = 20 \]

2. For \( n = 2 \):
\[ p = 2(2^4) + 3(2^3) + 4(2^2) + 5(2) + 3 = 2(16) + 3(8) + 4(4) + 5(2) + 3 = 32 + 24 + 16 + 10 + 3 = 85 \]

3. For \( n = 3 \):
\[ p = 2(3^4) + 3(3^3) + 4(3^2) + 5(3) + 2 = 2(81) + 3(27) + 4(9) + 5(3) + 2 = 162 + 81 + 36 + 15 + 2 = 296 \]

4. For \( n = 6 \):
\[ p = 2(6^4) + 3(6^3) + 4(6^2) + 5(6) + 1 = 2(1296) + 3(216) + 4(36) + 5(6) + 1 = 2592 + 648 + 144 + 30 + 1 = 3415 \]

### Step 5: Identify Unique Values of \( p \)

The values of \( p \) we calculated are:
- For \( n = 1 \): \( p = 20 \)
- For \( n = 2 \): \( p = 85 \)
- For \( n = 3 \): \( p = 296 \)
- For \( n = 6 \): \( p = 3415 \)

### Conclusion: Count the Number of Unique Elements in Set \( A \)

The unique values of \( p \) are \( 20, 85, 296, \) and \( 3415 \). Therefore, the number of elements in the set \( A \) is:

\[ \text{Number of elements in set } A = 4 \]

### Final Answer

The number of elements in the set \( A \) is \( 4 \).

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