How many `(i)` maximum `(ii)` minimum number of zeroes can a quadratic polynomial have ?

Answer

To determine the maximum and minimum number of zeros that a quadratic polynomial can have, we need to understand the nature of quadratic polynomials.

### Step-by-step Solution:

1. **Definition of a Quadratic Polynomial**:
A quadratic polynomial is generally expressed in the form:
\[ f(x) = ax^2 + bx + c \]
where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\). The degree of this polynomial is 2.

**Hint**: Remember that the degree of a polynomial indicates the highest power of the variable in the polynomial.

2. **Understanding the Relationship Between Degree and Zeros**:
The Fundamental Theorem of Algebra states that a polynomial of degree \(n\) can have at most \(n\) roots (or zeros). Therefore, for a quadratic polynomial (degree 2), it can have at most 2 zeros.

**Hint**: The maximum number of zeros is equal to the degree of the polynomial.

3. **Finding the Minimum Number of Zeros**:
A quadratic polynomial can also have fewer than 2 zeros. Specifically:
- It can have 2 distinct real zeros (roots).
- It can have 1 real zero (a repeated root).
- It can have no real zeros (but 2 complex zeros).

Thus, the minimum number of zeros a quadratic polynomial can have is 0.

**Hint**: Consider the nature of the roots: they can be real or complex, and sometimes they can repeat.

4. **Conclusion**:
- **Maximum number of zeros**: 2
- **Minimum number of zeros**: 0

### Final Answer:
(i) Maximum number of zeros: **2**
(ii) Minimum number of zeros: **0**