A nonzero polynomial with real coefficient has the property that `f(x)=f^(prime)(x)dotf^(prime)(x)dot` If `a` is the leading coefficient of `f(x),` then the value of `1//2a)` is____

Options

A. `1//3`
B. 6
C. 12
D. `1//18`

Explanation

To solve the problem step by step, we need to analyze the given polynomial function and its derivatives.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a polynomial \( f(x) \) such that:
\[ f(x) = f'(x) \cdot f''(x) \]
where \( f'(x) \) is the first derivative and \( f''(x) \) is the second derivative of \( f(x) \).

2. **Determine the Order of the Polynomial**:
Let the degree (order) of the polynomial \( f(x) \) be \( n \).
- The degree of \( f'(x) \) will be \( n - 1 \).
- The degree of \( f''(x) \) will be \( n - 2 \).
- Therefore, the degree of the product \( f'(x) \cdot f''(x) \) will be:
\[ (n - 1) + (n - 2) = 2n - 3 \]

3. **Equating the Degrees**:
Since \( f(x) \) and \( f'(x) \cdot f''(x) \) are equal, their degrees must also be equal:
\[ n = 2n - 3 \]
Rearranging gives:
\[ n - 2n = -3 \implies -n = -3 \implies n = 3 \]
Thus, the polynomial \( f(x) \) is of degree 3.

4. **General Form of the Polynomial**:
The general form of a cubic polynomial is:
\[ f(x) = ax^3 + bx^2 + cx + d \]

5. **Finding the Derivatives**:
- The first derivative \( f'(x) \) is:
\[ f'(x) = 3ax^2 + 2bx + c \]
- The second derivative \( f''(x) \) is:
\[ f''(x) = 6ax + 2b \]

6. **Substituting into the Given Equation**:
Substitute \( f'(x) \) and \( f''(x) \) into the equation:
\[ ax^3 + bx^2 + cx + d = (3ax^2 + 2bx + c)(6ax + 2b) \]

7. **Expanding the Right Side**:
We need to find the coefficient of \( x^3 \) on the right side:
\[ (3ax^2)(6ax) = 18a^2x^3 \]
The highest degree term on the left side is \( ax^3 \).

8. **Setting the Coefficients Equal**:
From the equality of the coefficients of \( x^3 \):
\[ a = 18a^2 \]
Rearranging gives:
\[ 18a^2 - a = 0 \implies a(18a - 1) = 0 \]
Since \( a \) is nonzero, we have:
\[ 18a - 1 = 0 \implies a = \frac{1}{18} \]

9. **Finding the Value of \( \frac{1}{2a} \)**:
Now, we need to find:
\[ \frac{1}{2a} = \frac{1}{2 \cdot \frac{1}{18}} = \frac{18}{2} = 9 \]

### Final Answer:
The value of \( \frac{1}{2a} \) is \( 9 \).