If `A={n:(n^(3)+5n^(2)+2)/(n)"is an integer"},`then the number of elements in the set A, is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to determine the values of \( n \) such that the expression

\[ \frac{n^3 + 5n^2 + 2}{n} \]

is an integer.

### Step 1: Simplify the expression

We start by simplifying the expression:

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

### Step 2: Identify conditions for integrality

For the entire expression \( n^2 + 5n + \frac{2}{n} \) to be an integer, the term \( \frac{2}{n} \) must also be an integer. This means that \( n \) must be a divisor of 2.

### Step 3: Find the divisors of 2

The divisors of 2 are:

\[ \pm 1, \pm 2 \]

### Step 4: List possible values of \( n \)

Thus, the possible integer values for \( n \) are:

\[ n = -2, -1, 1, 2 \]

### Step 5: Count the number of elements in set \( A \)

The set \( A \) consists of these values of \( n \):

\[ A = \{-2, -1, 1, 2\} \]

The number of elements in set \( A \) is:

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

### Final Answer

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

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