The number of real roots of the equation
`1+a_(1)x+a_(2)x^(2)+………..a_(n)x^(n)=0`, where `|x| lt (1)/(3)` and `|a_(n)| lt 2`, is

Options

A. n if n is even
B. 1 if n is odd
C. 0 for any ` n in N`
D. None of these

Answer

To solve the equation

\[ 1 + a_1 x + a_2 x^2 + \ldots + a_n x^n = 0 \]

given the conditions \( |x| < \frac{1}{3} \) and \( |a_n| < 2 \), we will analyze the polynomial and its roots step by step.

### Step 1: Understanding the Polynomial
We start with the polynomial:

\[ P(x) = 1 + a_1 x + a_2 x^2 + \ldots + a_n x^n \]

We need to find the number of real roots of this polynomial under the given constraints.

### Step 2: Analyzing the Constraints
We know that:

1. \( |x| < \frac{1}{3} \) implies that \( x \) is restricted to the interval \( (-\frac{1}{3}, \frac{1}{3}) \).
2. \( |a_n| < 2 \) indicates that the leading coefficient \( a_n \) is bounded.

### Step 3: Evaluating the Polynomial at the Boundaries
Let’s evaluate the polynomial at the boundaries of the interval \( x = -\frac{1}{3} \) and \( x = \frac{1}{3} \):

1. **At \( x = -\frac{1}{3} \)**:

\[ P\left(-\frac{1}{3}\right) = 1 + a_1\left(-\frac{1}{3}\right) + a_2\left(-\frac{1}{3}\right)^2 + \ldots + a_n\left(-\frac{1}{3}\right)^n \]

This value will depend on the coefficients \( a_1, a_2, \ldots, a_n \).

2. **At \( x = \frac{1}{3} \)**:

\[ P\left(\frac{1}{3}\right) = 1 + a_1\left(\frac{1}{3}\right) + a_2\left(\frac{1}{3}\right)^2 + \ldots + a_n\left(\frac{1}{3}\right)^n \]

Similarly, this value will also depend on the coefficients.

### Step 4: Finding the Values of the Polynomial
Since \( |a_n| < 2 \), we can say that the polynomial will not reach zero within the interval \( (-\frac{1}{3}, \frac{1}{3}) \) because:

- The polynomial \( P(x) \) is continuous.
- The values at the boundaries will not allow the polynomial to cross the x-axis.

### Step 5: Conclusion
Since the polynomial does not change sign in the interval \( (-\frac{1}{3}, \frac{1}{3}) \) and does not equal zero at the boundaries, we conclude that there are no real roots within the specified range.

Thus, the number of real roots of the equation is:

\[ \text{Number of real roots} = 0 \]

### Final Answer
The number of real roots of the equation is **0**.

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