If one zero of the polynomial `(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 solve the problem step by step, we will follow the logic provided in the video transcript.

### Step 1: Understand the problem
We are given a polynomial \(3x^2 + 8x + k\) and we know that one zero of this polynomial is the reciprocal of the other zero. We need to find the value of \(k\).

### Step 2: Set up the roots
Let the roots of the polynomial be \(\alpha\) and \(\frac{1}{\alpha}\), since one root is the reciprocal of the other.

### Step 3: Use the product of the roots
For a quadratic polynomial of the form \(ax^2 + bx + c\), the product of the roots can be expressed as:
\[ \text{Product of roots} = \frac{c}{a} \]
In our case, the polynomial is \(3x^2 + 8x + k\), where \(a = 3\), \(b = 8\), and \(c = k\).

### Step 4: Calculate the product of the roots
The product of the roots \(\alpha\) and \(\frac{1}{\alpha}\) is:
\[ \alpha \cdot \frac{1}{\alpha} = 1 \]
According to the formula, we also have:
\[ \frac{k}{3} \]
Setting these two expressions equal gives us:
\[ \frac{k}{3} = 1 \]

### Step 5: Solve for \(k\)
To find \(k\), we multiply both sides of the equation by 3:
\[ k = 3 \cdot 1 = 3 \]

### Final Answer
Thus, the value of \(k\) is \(3\).

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