The equation ` x^4 + 3x^3 -3x -1=0` is a reciprocal, equation of_______

Options

A. class one and odd order
B. class two and even order
C. class one and even order
D. class two and odd order

Answer

To determine the reciprocal equation of the given polynomial equation \( x^4 + 3x^3 - 3x - 1 = 0 \), we will follow these steps:

### Step 1: Understand the concept of reciprocal equations
A reciprocal equation is one where if \( x \) is a root, then \( \frac{1}{x} \) is also a root. For a polynomial equation of degree \( n \), the reciprocal equation will also be of degree \( n \).

### Step 2: Write the given equation
The given polynomial equation is:
\[ x^4 + 3x^3 - 3x - 1 = 0 \]

### Step 3: Replace \( x \) with \( \frac{1}{x} \)
To find the reciprocal equation, we substitute \( x \) with \( \frac{1}{x} \) in the original equation:
\[ \left(\frac{1}{x}\right)^4 + 3\left(\frac{1}{x}\right)^3 - 3\left(\frac{1}{x}\right) - 1 = 0 \]

### Step 4: Simplify the equation
Multiplying through by \( x^4 \) to eliminate the fractions gives:
\[ 1 + 3x - 3x^3 - x^4 = 0 \]
Rearranging this results in:
\[ -x^4 - 3x^3 + 3x + 1 = 0 \]
or equivalently:
\[ x^4 + 3x^3 - 3x - 1 = 0 \]

### Step 5: Identify the type of the reciprocal equation
Since the original equation and the resulting equation after substitution are the same, we can conclude that the reciprocal equation is of the same degree (4) and has the same roots.

### Step 6: Determine the classification
The classification of the equation is based on its degree and order. The degree is 4, and since it can be expressed in terms of \( x \) and \( \frac{1}{x} \), it is a reciprocal equation of class 2 and order 2.

### Conclusion
Thus, the equation \( x^4 + 3x^3 - 3x - 1 = 0 \) is a reciprocal equation of **class 2 and order 2**.

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