a quadratic polynomial , the sum of of whose zeroes is 0 and one zero is 3 is

The general form of a quadratic polynomial is:

\[
P(x) = ax^2 + bx + c
\]

According to Vieta's formulas:
- The sum of the zeroes of the quadratic polynomial is given by \(-\frac{b}{a}\).
- The product of the zeroes is \(\frac{c}{a}\).

Given that:
1. The sum of the zeroes is 0.
2. One of the zeroes is 3.

Let's call the other zero \(r\). Based on the first condition:

\[
3 + r = 0
\]

So, \(r = -3\).

Thus, the two zeroes are 3 and \(-3\).

Now, the quadratic polynomial can be written as:

\[
P(x) = a(x - 3)(x + 3)
\]

Simplifying this:

\[
P(x) = a(x^2 - 9)
\]

Since no specific value for \(a\) is given, the polynomial could be \(P(x) = a(x^2 - 9)\), where \(a\) is any constant.

If we assume \(a = 1\), the polynomial becomes:

\[
P(x) = x^2 - 9
\]

Would you like more details or have any questions?

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Here are some related questions you might find helpful:
1. What are Vieta's formulas for a quadratic equation?
2. How can you find the product of the zeroes of a quadratic polynomial?
3. What does the coefficient \(a\) represent in a quadratic equation?
4. How can you determine the discriminant of a quadratic equation?
5. Can a quadratic polynomial have zero as a root?

**Tip:** For any quadratic equation, the sum and product of the roots can be found using the coefficients of \(x\) and the constant term!

A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, is:

Discover how to determine a quadratic polynomial where the sum of the zeroes is 0 and one of the zeroes is 3. This solution explores algebraic concepts such as Vieta's formulas and the relationship between zeroes and coefficients. Perfect for students in Grades 8-10.

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