A quadratic polynomial with zeroes -2 and 3, is:

Options

A. `3x^2 - 2x + 6`
B. `2x^2 + 3x - 6 `
C. `x^2 -2x + 6`
D. `x^2 -x - 6`

Answer

To find a quadratic polynomial with given zeroes, we can follow these steps:

### Step 1: Identify the Zeroes
The zeroes of the quadratic polynomial are given as -2 and 3. Let's denote these zeroes as:
- α = -2
- β = 3

### Step 2: Use the Relationship Between Zeroes and Coefficients
For a quadratic polynomial in the standard form \( f(x) = ax^2 + bx + c \), the relationships between the zeroes and the coefficients are:
- Sum of the zeroes (α + β) = -b/a
- Product of the zeroes (αβ) = c/a

### Step 3: Calculate the Sum and Product of the Zeroes
Now, let's calculate the sum and product of the zeroes:
- Sum (α + β) = -2 + 3 = 1
- Product (αβ) = (-2) * 3 = -6

### Step 4: Write the Quadratic Polynomial
Using the relationships derived from the zeroes:
- The quadratic polynomial can be expressed as:
\[ f(x) = a(x^2 - (α + β)x + αβ) \]
Substituting the values we found:
\[ f(x) = a(x^2 - 1x - 6) \]
Assuming \( a = 1 \) (the simplest case), we have:
\[ f(x) = x^2 - x - 6 \]

### Step 5: Final Result
Thus, the required quadratic polynomial is:
\[ f(x) = x^2 - x - 6 \]