The number of positive integral roots of `x^(4) + x^(3) - 4 x^(2) + x + 1 = 0`, is

Options

A. 0
B. 1
C. 2
D. 4

Answer

To find the number of positive integral roots of the polynomial equation \( f(x) = x^4 + x^3 - 4x^2 + x + 1 = 0 \), we will use Descartes' Rule of Signs. This rule helps us determine the number of positive and negative roots by counting the number of sign changes in the polynomial.

### Step-by-Step Solution:

1. **Define the function**:
We start by defining the polynomial function:
\[ f(x) = x^4 + x^3 - 4x^2 + x + 1 \]

2. **Evaluate the signs of the coefficients**:
We will look at the coefficients of the polynomial:
- The coefficient of \( x^4 \) is positive (+1).
- The coefficient of \( x^3 \) is positive (+1).
- The coefficient of \( x^2 \) is negative (-4).
- The coefficient of \( x^1 \) is positive (+1).
- The constant term is positive (+1).

This gives us the sequence of signs: \( +, +, -, +, + \).

3. **Count the sign changes**:
Now we will count the number of sign changes in this sequence:
- From \( + \) to \( + \): no change.
- From \( + \) to \( - \): 1 change.
- From \( - \) to \( + \): 1 change.
- From \( + \) to \( + \): no change.

Total number of sign changes = 2.

4. **Apply Descartes' Rule of Signs**:
According to Descartes' Rule of Signs, the number of positive roots is equal to the number of sign changes or less than that by an even number. Since we have 2 sign changes, the possible number of positive integral roots is 2 or 0.

5. **Conclusion**:
Since we are looking for positive integral roots, we conclude that the number of positive integral roots of the polynomial \( f(x) = 0 \) is 2.

### Final Answer:
The number of positive integral roots of the equation \( x^4 + x^3 - 4x^2 + x + 1 = 0 \) is **2**.