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 can follow these steps:

### Step 1: Identify the zeroes
The zeroes given are:
- α = 1
- β = -2

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

### Step 3: Calculate the product of the zeroes
The product of the zeroes (α * β) is calculated as follows:
\[ \text{Product of zeroes} = α \cdot β = 1 \cdot (-2) = -2 \]

### Step 4: Write the quadratic polynomial
Using the standard form of a quadratic polynomial based on its zeroes, we can express it as:
\[ p(x) = x^2 - (\text{Sum of zeroes}) \cdot x + (\text{Product of zeroes}) \]
Substituting the values we found:
\[ p(x) = x^2 - (-1) \cdot x + (-2) \]
This simplifies to:
\[ p(x) = x^2 + x - 2 \]

### Step 5: Final form of the polynomial
Thus, the quadratic polynomial whose zeroes are 1 and -2 is:
\[ \boxed{x^2 + x - 2} \]

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