find the quadratic polynomial the sum and product of whose zeroes are -3 and 2 respectively

The general form of a quadratic polynomial when the sum and product of its zeroes are given can be written as:

\[
p(x) = x^2 - (\text{sum of zeroes})x + (\text{product of zeroes})
\]

Here, the given sum of the zeroes is \(-3\) and the product of the zeroes is \(2\). Substituting these values into the formula:

\[
p(x) = x^2 - (-3)x + 2
\]

Simplify:

\[
p(x) = x^2 + 3x + 2
\]

Thus, the quadratic polynomial is:

\[
\boxed{p(x) = x^2 + 3x + 2}
\]

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Would you like me to explain further or verify by solving for the roots?  
Here are five related questions to explore:  

1. How do you verify that given roots satisfy the polynomial?  
2. Can you derive the quadratic equation if roots are given explicitly instead of their sum and product?  
3. What is the relationship between coefficients and roots in any polynomial?  
4. How do you factorize a quadratic polynomial like \(x^2 + 3x + 2\)?  
5. What is the graphical interpretation of the sum and product of the roots?

**Tip:** The coefficient of \(x\) in a quadratic equation is always \(-(\text{sum of the roots})\).

Discover how to determine the quadratic polynomial when the sum of its zeroes is -3 and the product of its zeroes is 2. This solution explores the relationship between roots and coefficients using algebraic concepts, ideal for students in Grades 8-10.

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