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

Options

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

Answer

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

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

### Step 2: Calculate the sum of the zeroes
The sum of the zeroes \( S \) can be calculated as:
\[ S = \alpha + \beta = -3 + 4 = 1 \]

### Step 3: Calculate the product of the zeroes
The product of the zeroes \( P \) can be calculated as:
\[ P = \alpha \times \beta = -3 \times 4 = -12 \]

### Step 4: Write the general form of the quadratic polynomial
The general form of a quadratic polynomial with zeroes \( \alpha \) and \( \beta \) is given by:
\[ f(x) = x^2 - Sx + P \]
Substituting the values of \( S \) and \( P \):
\[ f(x) = x^2 - (1)x + (-12) \]
This simplifies to:
\[ f(x) = x^2 - x - 12 \]

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