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 the quadratic polynomial with zeroes -2 and 3, we can follow these steps:

### Step 1: Identify the zeroes
The zeroes of the polynomial are given as:
- α = -2
- β = 3

### Step 2: Write the polynomial in factored form
The quadratic polynomial can be expressed in the factored form as:
\[ P(x) = (x - α)(x - β) \]
Substituting the values of α and β:
\[ P(x) = (x - (-2))(x - 3) \]
This simplifies to:
\[ P(x) = (x + 2)(x - 3) \]

### Step 3: Expand the polynomial
Now, we will expand the expression:
\[ P(x) = (x + 2)(x - 3) \]
Using the distributive property (FOIL method):
\[ P(x) = x \cdot x + x \cdot (-3) + 2 \cdot x + 2 \cdot (-3) \]
This simplifies to:
\[ P(x) = x^2 - 3x + 2x - 6 \]

### Step 4: Combine like terms
Now, combine the like terms:
\[ P(x) = x^2 + (-3x + 2x) - 6 \]
This simplifies to:
\[ P(x) = x^2 - x - 6 \]

### Final Answer
Thus, the quadratic polynomial with zeroes -2 and 3 is:
\[ P(x) = x^2 - x - 6 \]

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