Write the quadratic polynomial, whose zeroes and 1 and -2.

Options

A. `x^2+x+2`
B. `x^2+x-2`
C. `x^2-x-2`
D. `x^2+2x-1`

Answer

To find the quadratic polynomial whose zeroes are 1 and -2, we will follow these steps:

### Step 1: Identify the zeroes
The zeroes (roots) of the polynomial are given as:
- \( \alpha = 1 \)
- \( \beta = -2 \)

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

### Step 3: Calculate the product of the zeroes
The product of the zeroes \( P \) is calculated as follows:
\[ P = \alpha \cdot \beta = 1 \cdot (-2) = -2 \]

### Step 4: Form the quadratic polynomial
Using the standard form of a quadratic polynomial based on its zeroes:
\[ f(x) = x^2 - (S)x + P \]
Substituting the values of \( S \) and \( P \):
\[ f(x) = x^2 - (-1)x + (-2) \]
This simplifies to:
\[ f(x) = x^2 + x - 2 \]

### Final Answer
The quadratic polynomial whose zeroes are 1 and -2 is:
\[ f(x) = x^2 + x - 2 \]

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