If one zero of the polynomal `(3x^(2) + 8x + k)` is the reciprocal of the other, then value of k is

Options

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

Answer

To find the value of \( k \) in the polynomial \( 3x^2 + 8x + k \) given that one zero is the reciprocal of the other, we can follow these steps:

### Step 1: Understand the relationship between the zeros
Let the two zeros of the polynomial be \( \alpha \) and \( \beta \). According to the problem, if one zero is the reciprocal of the other, we can express this relationship as:
\[ \beta = \frac{1}{\alpha} \]

### Step 2: Use the product of the zeros
The product of the zeros of a quadratic polynomial \( ax^2 + bx + c \) is given by:
\[ \alpha \cdot \beta = \frac{c}{a} \]
In our case, \( a = 3 \), \( b = 8 \), and \( c = k \). Therefore, we can write:
\[ \alpha \cdot \beta = \frac{k}{3} \]

### Step 3: Substitute the relationship of the zeros
Since \( \beta = \frac{1}{\alpha} \), we can substitute this into the product of the zeros:
\[ \alpha \cdot \frac{1}{\alpha} = 1 \]
Thus, we have:
\[ 1 = \frac{k}{3} \]

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

### Conclusion
The value of \( k \) is \( 3 \).

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