Understanding Quadratic Polynomials
A quadratic polynomial generally takes the form:
\[ P(x) = ax^2 + bx + c \]
Where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).

Sum and Product of Zeroes
For a quadratic polynomial, the sum and product of its zeroes can be derived from the coefficients:
- **Sum of Zeroes** (\( \alpha + \beta \)) is given by \( -\frac{b}{a} \)
- **Product of Zeroes** (\( \alpha \cdot \beta \)) is given by \( \frac{c}{a} \)
Given:
- Sum of Zeroes = 5
- Product of Zeroes = 0

Setting Up the Polynomial
From the above information, we can set up the relationships:
- \( -\frac{b}{a} = 5 \) (1)
- \( \frac{c}{a} = 0 \) (2)

Deriving Coefficients
From equation (2), since the product of the zeroes is 0, it implies that at least one zero must be 0. If \( c = 0 \), we can simplify our polynomial as follows:
1. Let \( a = 1 \) for simplicity (you can choose any non-zero value).
2. From (1), substituting \( a = 1 \):
- \( -b = 5 \)
- Therefore, \( b = -5 \).

Forming the Polynomial
Now substituting \( a \) and \( b \) back into the polynomial, we get:
\[ P(x) = 1x^2 - 5x + 0 \]
or simply:
\[ P(x) = x^2 - 5x \]

Conclusion
Thus, the quadratic polynomial with the specified sum and product of its zeroes is:
\[ P(x) = x^2 - 5x \]
This polynomial has zeroes at \( 0 \) and \( 5 \).