Let `P_n(ix) =1+2x+3x^2+............+(n+1)x^n` be a polynomial such that n is even.The number of real roots of `P_n(x)=0` is

Options

A. 0
B. n
C. 1
D. n+1

Answer

To solve the problem, we need to analyze the polynomial \( P_n(x) = 1 + 2x + 3x^2 + \ldots + (n+1)x^n \) where \( n \) is even, and determine the number of real roots of the equation \( P_n(x) = 0 \).

### Step 1: Understand the Polynomial
The polynomial \( P_n(x) \) is defined as:
\[ P_n(x) = 1 + 2x + 3x^2 + \ldots + (n+1)x^n \]
where \( n \) is an even integer.

### Step 2: Evaluate the Polynomial at Specific Points
Let's evaluate \( P_n(x) \) at \( x = 0 \):
\[ P_n(0) = 1 + 2(0) + 3(0)^2 + \ldots + (n+1)(0)^n = 1 \]
This shows that \( P_n(0) = 1 \), which is positive.

### Step 3: Analyze the Behavior of the Polynomial for \( x > 0 \)
For \( x > 0 \), all terms in the polynomial are positive:
\[ P_n(x) = 1 + 2x + 3x^2 + \ldots + (n+1)x^n > 0 \]
Thus, \( P_n(x) > 0 \) for all \( x > 0 \).

### Step 4: Analyze the Behavior of the Polynomial for \( x < 0 \)
Now, consider \( x < 0 \). Since \( n \) is even, the leading term \( (n+1)x^n \) will be positive as \( x^n \) will be positive when \( n \) is even. The polynomial can be expressed as:
\[ P_n(x) = 1 + 2x + 3x^2 + \ldots + (n+1)x^n \]
As \( x \) approaches negative infinity, the leading term dominates:
\[ P_n(x) \to +\infty \text{ as } x \to -\infty \]

### Step 5: Check for Roots in the Interval \( (-\infty, 0) \)
To find the number of real roots, we need to check if \( P_n(x) \) can cross the x-axis in the interval \( (-\infty, 0) \). Since \( P_n(0) = 1 > 0 \) and \( P_n(x) \to +\infty \) as \( x \to -\infty \), we need to check if there are any points where \( P_n(x) \) could be negative.

### Step 6: Conclusion
Since \( P_n(x) \) is continuous and starts at a positive value at \( x = 0 \) and goes to positive infinity as \( x \to -\infty \), and because all terms contribute positively for \( x > 0 \), we conclude that:
- \( P_n(x) \) does not cross the x-axis.
- Therefore, there are no real roots.

Thus, the number of real roots of \( P_n(x) = 0 \) is:
\[ \boxed{0} \]