A quadratic polynomial, whose zeroes are -3 and 4, is

Options

A. `x^(2)-x+12`
B. `x^(2)+x+12`
C. `(x^(2))/2-x/2-6`
D. `2x^(2)+2x-24`

Answer

To find a quadratic polynomial whose zeroes are -3 and 4, we can follow these steps:

### Step 1: Identify the zeroes
The zeroes given are:
- \( \alpha = -3 \)
- \( \beta = 4 \)

### Step 2: Calculate the sum of the zeroes
The sum of the zeroes \( \alpha + \beta \) is calculated as follows:
\[ \text{Sum of zeroes} = -3 + 4 = 1 \]

### Step 3: Calculate the product of the zeroes
The product of the zeroes \( \alpha \cdot \beta \) is calculated as follows:
\[ \text{Product of zeroes} = -3 \times 4 = -12 \]

### Step 4: Write the quadratic polynomial
Using the standard form of a quadratic polynomial, which is given by:
\[ P(x) = x^2 - (\text{Sum of zeroes}) \cdot x + (\text{Product of zeroes}) \]
we can substitute the values we calculated:
\[ P(x) = x^2 - (1)x + (-12) \]
This simplifies to:
\[ P(x) = x^2 - x - 12 \]

### Step 5: Final polynomial
Thus, the quadratic polynomial whose zeroes are -3 and 4 is:
\[ P(x) = x^2 - x - 12 \]