The equation ` 6x ^5 + 7x^4 +12x^3+ 12x^2 + 7x +6=0` is a reciprocal equation of

Options

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

Answer

To determine the class and order of the given equation \( 6x^5 + 7x^4 + 12x^3 + 12x^2 + 7x + 6 = 0 \), we will analyze the coefficients of the polynomial.

### Step 1: Identify the coefficients of the polynomial
The given polynomial is:
\[ 6x^5 + 7x^4 + 12x^3 + 12x^2 + 7x + 6 \]
The coefficients are:
- Coefficient of \( x^5 \): 6
- Coefficient of \( x^4 \): 7
- Coefficient of \( x^3 \): 12
- Coefficient of \( x^2 \): 12
- Coefficient of \( x^1 \): 7
- Constant term (coefficient of \( x^0 \)): 6

### Step 2: Check for reciprocal properties
For a polynomial to be classified as a reciprocal polynomial, the coefficients must satisfy the following conditions:
- The coefficient of \( x^n \) (where \( n \) is the degree of the polynomial) must be equal to the constant term.
- The coefficient of \( x^{n-1} \) must be equal to the coefficient of \( x^1 \).
- The coefficient of \( x^{n-2} \) must be equal to the coefficient of \( x^2 \).
- And so on...

Let's check these conditions:
- Coefficient of \( x^5 \) (6) is equal to the constant term (6).
- Coefficient of \( x^4 \) (7) is equal to the coefficient of \( x^1 \) (7).
- Coefficient of \( x^3 \) (12) is equal to the coefficient of \( x^2 \) (12).

Since all these conditions are satisfied, the polynomial is indeed a reciprocal polynomial.

### Step 3: Determine the class and order
Next, we need to determine the class and order of the polynomial:
- **Class**: The polynomial is of class 1 because the coefficients match as described above.
- **Order**: The polynomial has no middle term (the term \( x^3 \) and \( x^2 \) are present but are equal), which indicates that it is of even order.

### Conclusion
Thus, the given polynomial \( 6x^5 + 7x^4 + 12x^3 + 12x^2 + 7x + 6 = 0 \) is a reciprocal equation of **class 1 and even order**.

### Final Answer
The correct option is: **Class 1 and Even Order**.

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