What is the value of k in the quadratic polynomial `kx^(2) + 4x + 3k`, if the sum of the zeroes is equal to their product?

Options

A. `-(4)/(3)`
B. `(2)/(3)`
C. `(1)/(0)`
D. `-5`

Answer

To find the value of \( k \) in the quadratic polynomial \( kx^2 + 4x + 3k \) such that the sum of the zeroes is equal to their product, we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic polynomial is in the form \( ax^2 + bx + c \), where:
- \( a = k \)
- \( b = 4 \)
- \( c = 3k \)

### Step 2: Write the formulas for the sum and product of the roots
For a quadratic polynomial \( ax^2 + bx + c \):
- The sum of the roots (zeroes) is given by \( -\frac{b}{a} \).
- The product of the roots is given by \( \frac{c}{a} \).

### Step 3: Calculate the sum and product of the roots
Using the coefficients identified:
- Sum of the roots \( S = -\frac{b}{a} = -\frac{4}{k} \)
- Product of the roots \( P = \frac{c}{a} = \frac{3k}{k} = 3 \)

### Step 4: Set the sum equal to the product
According to the problem, the sum of the zeroes is equal to their product:
\[ -\frac{4}{k} = 3 \]

### Step 5: Solve for \( k \)
To solve for \( k \), we first multiply both sides by \( k \) (assuming \( k \neq 0 \)):
\[ -4 = 3k \]
Now, divide both sides by 3:
\[ k = -\frac{4}{3} \]

### Conclusion
The value of \( k \) is:
\[ \boxed{-\frac{4}{3}} \]