Find the quadratic polynomial, whose zeroes are -3 and 4.

Options

A. `x^2-x-12`
B. `2x^2-x+12`
C. `x^2-x+4`
D. `x^2-3x+4`

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 are:
- α = -3
- β = 4

### Step 2: Write the polynomial in factored form
A quadratic polynomial can be expressed in the factored form as:
\[ f(x) = (x - α)(x - β) \]

Substituting the values of α and β, we have:
\[ f(x) = (x - (-3))(x - 4) \]
This simplifies to:
\[ f(x) = (x + 3)(x - 4) \]

### Step 3: Expand the polynomial
Now, we will expand the expression:
\[ f(x) = (x + 3)(x - 4) \]

Using the distributive property (also known as the FOIL method for binomials):
\[ f(x) = x^2 - 4x + 3x - 12 \]

### Step 4: Combine like terms
Now, combine the like terms:
\[ f(x) = x^2 - 4x + 3x - 12 \]
\[ f(x) = x^2 - x - 12 \]

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

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