A quadratic polynomial whose zeroes are 3 and -5, is

Options

A. `x^(2) + 2x - 15`
B. `x^(2) - 2x + 15`
C. `x^(2) - 8x + 15`
D. `x^(2) + 8x - 15 `

Answer

To find a quadratic polynomial whose zeroes are 3 and -5, we can use the relationship between the zeroes and the coefficients of the polynomial.

### Step 1: Identify the zeroes
Let the zeroes of the quadratic polynomial be \( \alpha = 3 \) and \( \beta = -5 \).

### Step 2: Use the formula for the quadratic polynomial
The general form of a quadratic polynomial with zeroes \( \alpha \) and \( \beta \) can be expressed as:
\[ P(x) = x^2 - (\alpha + \beta)x + \alpha \beta \]

### Step 3: Calculate \( \alpha + \beta \)
Now, we calculate the sum of the zeroes:
\[ \alpha + \beta = 3 + (-5) = 3 - 5 = -2 \]

### Step 4: Calculate \( \alpha \beta \)
Next, we calculate the product of the zeroes:
\[ \alpha \beta = 3 \times (-5) = -15 \]

### Step 5: Substitute the values into the polynomial
Now, substitute \( \alpha + \beta \) and \( \alpha \beta \) back into the polynomial formula:
\[ P(x) = x^2 - (-2)x + (-15) \]
This simplifies to:
\[ P(x) = x^2 + 2x - 15 \]

### Step 6: Final polynomial
Thus, the quadratic polynomial whose zeroes are 3 and -5 is:
\[ P(x) = x^2 + 2x - 15 \]