If the equation whose roots are p times the roots of `x^4 +2x^3 +46 x^2 +8x+16 =0` is a reciprocal equation then p=

Options

A. 2
B. 3
C. `+- 1/2`
D. `+-1/3`

Answer

To solve the problem, we need to find the value of \( p \) such that the equation whose roots are \( p \) times the roots of the polynomial \( x^4 + 2x^3 + 46x^2 + 8x + 16 = 0 \) is a reciprocal equation.

### Step-by-Step Solution:

1. **Identify the given polynomial and its roots**:
The given polynomial is:
\[ x^4 + 2x^3 + 46x^2 + 8x + 16 = 0 \]
Let the roots of this polynomial be \( \alpha, \beta, \gamma, \delta \).

2. **Calculate the product of the roots**:
The product of the roots of a polynomial \( ax^n + bx^{n-1} + ... + z = 0 \) is given by:
\[ \text{Product of roots} = \frac{(-1)^n \cdot \text{constant term}}{a} \]
Here, \( n = 4 \) (degree of the polynomial), the constant term is \( 16 \), and \( a = 1 \) (coefficient of \( x^4 \)):
\[ \text{Product of roots} = \frac{(-1)^4 \cdot 16}{1} = 16 \]

3. **Define the new roots**:
According to the problem, the new roots are \( p\alpha, p\beta, p\gamma, p\delta \).

4. **Set up the product of the new roots**:
The product of the new roots can be expressed as:
\[ p^4 \cdot (\alpha \beta \gamma \delta) = p^4 \cdot 16 \]

5. **Condition for reciprocal equations**:
For the equation to be a reciprocal equation, the product of the new roots must equal 1:
\[ p^4 \cdot 16 = 1 \]

6. **Solve for \( p^4 \)**:
Rearranging the equation gives:
\[ p^4 = \frac{1}{16} \]

7. **Take the fourth root**:
Taking the fourth root of both sides, we get:
\[ p = \pm \frac{1}{2} \]

### Final Answer:
Thus, the values of \( p \) are:
\[ p = \pm \frac{1}{2} \]