The roots of the equation `3x^(4)+4x^(3)+x-1=0` consist of

Options

A. three positive real numbers and one negative real number
B. three negative real numbers and one positive real number
C. one negative real number and three imaginary numbers
D. one positive real number , one negative real number, and two imaginary numbers.

Explanation

To determine the nature of the roots of the polynomial equation \(3x^4 + 4x^3 + x - 1 = 0\), we will use Descartes' Rule of Signs. This rule helps us analyze the number of positive and negative roots based on the sign changes in the polynomial.

### Step 1: Analyze the polynomial for positive roots
The given polynomial is:
\[ f(x) = 3x^4 + 4x^3 + x - 1 \]
We will look for sign changes in \(f(x)\):
- The coefficients are: \(3\) (positive), \(4\) (positive), \(1\) (positive), and \(-1\) (negative).
- The sequence of signs is: \(+, +, +, -\).

**Sign changes**: There is **1 sign change** (from \(+\) to \(-\)).

According to Descartes' Rule of Signs:
- The number of positive roots is either equal to the number of sign changes or less than it by an even number. Therefore, the number of positive roots is **1**.

### Step 2: Analyze the polynomial for negative roots
Next, we will substitute \(x\) with \(-x\) to find the number of negative roots:
\[ f(-x) = 3(-x)^4 + 4(-x)^3 + (-x) - 1 = 3x^4 - 4x^3 - x - 1 \]
Now, we look for sign changes in \(f(-x)\):
- The coefficients are: \(3\) (positive), \(-4\) (negative), \(-1\) (negative), and \(-1\) (negative).
- The sequence of signs is: \(+, -, -, -\).

**Sign changes**: There is **1 sign change** (from \(+\) to \(-\)).

According to Descartes' Rule of Signs:
- The number of negative roots is also **1**.

### Step 3: Determine the number of imaginary roots
The total number of roots for a polynomial of degree 4 is 4. We have found:
- Number of positive roots = 1
- Number of negative roots = 1

Thus, the number of imaginary roots can be calculated as:
\[ \text{Number of imaginary roots} = \text{Total roots} - (\text{Number of positive roots} + \text{Number of negative roots}) \]
\[ \text{Number of imaginary roots} = 4 - (1 + 1) = 2 \]

### Conclusion
The roots of the polynomial \(3x^4 + 4x^3 + x - 1 = 0\) consist of:
- 1 positive real root
- 1 negative real root
- 2 imaginary roots

Thus, the correct option is:
**1 positive real number, 1 negative real number, and 2 imaginary numbers.**