If one root of the polynomial f(x) = `3x^(2)` + 11x + p is reciprocal of the other, then the value of p is

Options

A. 0
B. 3
C. `(1)/(3)`
D. `-3`

Answer

To find the value of \( p \) in the polynomial \( f(x) = 3x^2 + 11x + p \), given that one root is the reciprocal of the other, we can follow these steps:

### Step 1: Understand the roots
Let the roots of the polynomial be \( \alpha \) and \( \frac{1}{\alpha} \).

### Step 2: Use the sum of roots formula
For a quadratic polynomial \( ax^2 + bx + c = 0 \), the sum of the roots is given by:
\[ \text{Sum of roots} = -\frac{b}{a} \]
In our case, \( a = 3 \) and \( b = 11 \). Therefore, the sum of the roots can be calculated as:
\[ \alpha + \frac{1}{\alpha} = -\frac{11}{3} \]

### Step 3: Use the product of roots formula
The product of the roots for the polynomial is given by:
\[ \text{Product of roots} = \frac{c}{a} \]
Here, \( c = p \) and \( a = 3 \). Thus, we have:
\[ \alpha \cdot \frac{1}{\alpha} = \frac{p}{3} \]
Since \( \alpha \cdot \frac{1}{\alpha} = 1 \), we can set up the equation:
\[ 1 = \frac{p}{3} \]

### Step 4: Solve for \( p \)
To find \( p \), we can multiply both sides of the equation by 3:
\[ p = 3 \]

### Conclusion
Thus, the value of \( p \) is \( 3 \).

---