A quadratic polynomial whose one zero is `-5` and the product of the zeroes is 0 , is .

Options

A. `x^(2) + 5x`
B. `x^(2) - 5x`
C. `x^(2) + 5x + 5`
D. `x^(2) - 5x + 1 `

Answer

To find the quadratic polynomial given that one zero is \(-5\) and the product of the zeroes is \(0\), we can follow these steps:

### Step 1: Identify the roots
Let the roots of the quadratic polynomial be \(\alpha\) and \(\beta\). According to the problem, we know:
- One root, \(\alpha = -5\)
- The product of the roots, \(\alpha \cdot \beta = 0\)

### Step 2: Find the other root
Since the product of the roots is \(0\), we can set up the equation:
\[ \alpha \cdot \beta = -5 \cdot \beta = 0 \]
This implies that \(\beta\) must be \(0\) because the only way for the product to be \(0\) is if at least one of the roots is \(0\).

### Step 3: Calculate the sum of the roots
Now we have both roots:
- \(\alpha = -5\)
- \(\beta = 0\)

We can calculate the sum of the roots:
\[ \alpha + \beta = -5 + 0 = -5 \]

### Step 4: Write the quadratic polynomial
Using the standard form of a quadratic polynomial, which is given by:
\[ x^2 - (\text{sum of roots}) \cdot x + (\text{product of roots}) = 0 \]
Substituting the values we found:
- Sum of roots = \(-5\)
- Product of roots = \(0\)

The polynomial can be expressed as:
\[ x^2 - (-5)x + 0 = x^2 + 5x \]

### Step 5: Final form of the polynomial
Thus, the quadratic polynomial is:
\[ x^2 + 5x \]