Are the following statements True or False Justify your answers.
If the graph of a polynomial intersects the x- axis at exactly two points ,it need not be a quadratic polynomial .

Answer

To determine whether the statement "If the graph of a polynomial intersects the x-axis at exactly two points, it need not be a quadratic polynomial" is true or false, we can analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Statement**: The statement claims that a polynomial can intersect the x-axis at exactly two points without being a quadratic polynomial.

2. **Definition of Polynomial Degree**: A polynomial of degree \( n \) can have at most \( n \) real roots. A quadratic polynomial is a polynomial of degree 2, which can have up to 2 real roots.

3. **Graph of Quadratic Polynomial**: A quadratic polynomial can intersect the x-axis at two distinct points. For example, the polynomial \( f(x) = (x - 1)(x - 2) \) intersects the x-axis at \( x = 1 \) and \( x = 2 \).

4. **Higher Degree Polynomials**: Consider a polynomial of degree 4, such as \( f(x) = (x - 1)(x - 2)(x^2 + 1) \). This polynomial has two real roots (1 and 2) and two complex roots (the roots of \( x^2 + 1 \)). The graph of this polynomial will also intersect the x-axis at exactly two points (1 and 2).

5. **Conclusion**: Since we can have a polynomial of degree higher than 2 (like degree 4) that intersects the x-axis at exactly two points, the statement is true.

### Final Answer:
The statement is **True**. A polynomial can intersect the x-axis at exactly two points and still be of a degree higher than 2, such as a 4th degree polynomial.