Find the quadratic polynomial, the sum and product of whose zeroes are - 3 and 2 respectively. Hence find the zeroes.

Answer

To find the quadratic polynomial whose sum and product of zeroes are given, we can follow these steps:

### Step 1: Understand the relationship between the coefficients and the zeroes
For a quadratic polynomial of the form \( P(x) = ax^2 + bx + c \), the sum of the zeroes \( \alpha + \beta \) is given by \( -\frac{b}{a} \) and the product of the zeroes \( \alpha \beta \) is given by \( \frac{c}{a} \).

### Step 2: Set up the polynomial
Given:
- Sum of the zeroes \( \alpha + \beta = -3 \)
- Product of the zeroes \( \alpha \beta = 2 \)

We can use the standard form of the quadratic polynomial:
\[ P(x) = x^2 - (\text{sum of zeroes}) \cdot x + (\text{product of zeroes}) \]

Substituting the values:
\[ P(x) = x^2 - (-3)x + 2 \]
\[ P(x) = x^2 + 3x + 2 \]

### Step 3: Factor the polynomial
Next, we need to factor the polynomial \( P(x) = x^2 + 3x + 2 \).

To factor, we look for two numbers that multiply to \( 2 \) (the product of the zeroes) and add up to \( 3 \) (the sum of the zeroes). The numbers \( 1 \) and \( 2 \) satisfy this condition.

Thus, we can write:
\[ P(x) = (x + 1)(x + 2) \]

### Step 4: Find the zeroes
To find the zeroes, we set the polynomial equal to zero:
\[ (x + 1)(x + 2) = 0 \]

This gives us:
1. \( x + 1 = 0 \) → \( x = -1 \)
2. \( x + 2 = 0 \) → \( x = -2 \)

### Conclusion
The quadratic polynomial is \( P(x) = x^2 + 3x + 2 \), and the zeroes are \( x = -1 \) and \( x = -2 \).

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