A quadratic polynomial ,whose zeroes are `-3and4`, 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 of the polynomial are given as:
- α = -3
- β = 4

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

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

### Step 4: Write the standard form of the quadratic polynomial
The standard form of a quadratic polynomial with zeroes α and β is given by:
\[ P(x) = x^2 - (\text{Sum}) \cdot x + (\text{Product}) \]
Substituting the values we calculated:
\[ P(x) = x^2 - (1) \cdot x + (-12) \]
This simplifies to:
\[ P(x) = x^2 - x - 12 \]

### Step 5: Verify if the polynomial can be simplified or factored
We can check if the polynomial can be factored or simplified further. In this case, it cannot be factored easily, but we can express it in a different form if needed.

### Step 6: Final polynomial
The final quadratic polynomial whose zeroes are -3 and 4 is:
\[ P(x) = x^2 - x - 12 \]