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 set \( A = \{ n : \frac{n^3 + 5n^2 + 2}{n} \text{ is an integer} \} \).

### Step-by-Step Solution:

1. **Simplify the Expression**:
We start with the expression given in the set:
\[ \frac{n^3 + 5n^2 + 2}{n} \]
We can simplify this by dividing each term in the numerator by \( n \):
\[ = n^2 + 5n + \frac{2}{n} \]

2. **Identify Conditions for Integer**:
For the entire expression to be an integer, \( \frac{2}{n} \) must also be an integer. This means that \( n \) must be a divisor of 2.

3. **Find Divisors of 2**:
The divisors of 2 are:
\[ n = \pm 1, \pm 2 \]
These are the integers that can divide 2 without leaving a remainder.

4. **Check Each Divisor**:
Now we check each divisor to see if it satisfies the condition:
- For \( n = 1 \):
\[ n^2 + 5n + \frac{2}{n} = 1^2 + 5 \cdot 1 + \frac{2}{1} = 1 + 5 + 2 = 8 \quad (\text{integer}) \]
- For \( n = -1 \):
\[ n^2 + 5n + \frac{2}{n} = (-1)^2 + 5 \cdot (-1) + \frac{2}{-1} = 1 - 5 - 2 = -6 \quad (\text{integer}) \]
- For \( n = 2 \):
\[ n^2 + 5n + \frac{2}{n} = 2^2 + 5 \cdot 2 + \frac{2}{2} = 4 + 10 + 1 = 15 \quad (\text{integer}) \]
- For \( n = -2 \):
\[ n^2 + 5n + \frac{2}{n} = (-2)^2 + 5 \cdot (-2) + \frac{2}{-2} = 4 - 10 - 1 = -7 \quad (\text{integer}) \]

5. **Conclusion**:
All four values \( n = -2, -1, 1, 2 \) yield integers when substituted into the expression. Therefore, the set \( A \) contains these four elements.

Thus, the number of elements in the set \( A \) is:
\[ \text{Number of elements in } A = 4 \]