Out of the following, the incorrect statement for a quadratic polynomial is:

Options

A. no real zeroes
B. two equal real zeroes
C. two distinct zeroes
D. three real zeroes

Answer

To determine the incorrect statement regarding quadratic polynomials, we need to analyze the properties of quadratic polynomials and their zeros. A quadratic polynomial is generally expressed in the form \( ax^2 + bx + c \), where \( a \neq 0 \).

### Step-by-Step Solution:

1. **Understanding Quadratic Polynomials**:
A quadratic polynomial can have different types of zeros based on its discriminant (\( D \)), which is calculated as \( D = b^2 - 4ac \).

2. **Analyzing the Statements**:
We need to evaluate the following statements regarding the zeros of quadratic polynomials:
- **Statement 1**: A quadratic polynomial can have no real zeros.
- **Statement 2**: A quadratic polynomial can have equal real zeros.
- **Statement 3**: A quadratic polynomial can have two distinct real zeros.
- **Statement 4**: A quadratic polynomial can have three real zeros.

3. **Evaluating Each Statement**:
- **Statement 1**: A quadratic polynomial can have no real zeros if the discriminant \( D < 0 \). For example, the polynomial \( x^2 + 1 \) has no real zeros since \( D = 0^2 - 4(1)(1) = -4 \).
- **Statement 2**: A quadratic polynomial can have equal real zeros if the discriminant \( D = 0 \). For example, \( x^2 - 2x + 1 = (x-1)^2 \) has a double root at \( x = 1 \).
- **Statement 3**: A quadratic polynomial can have two distinct real zeros if the discriminant \( D > 0 \). For example, \( x^2 - 3x + 2 = (x-1)(x-2) \) has distinct zeros at \( x = 1 \) and \( x = 2 \).
- **Statement 4**: A quadratic polynomial cannot have three real zeros. A quadratic polynomial can have at most two zeros, as it is a second-degree polynomial.

4. **Conclusion**:
The incorrect statement is **Statement 4**, which claims that a quadratic polynomial can have three real zeros. This is not possible.

### Final Answer:
The incorrect statement for a quadratic polynomial is: **A quadratic polynomial can have three real zeros.**

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