Are the following statements True or False Justify your answers.
If the graph of a polynomial intersects the x- axis at only one point ,it cannot be a quadratic polynomial .

Answer

To determine whether the statement "If the graph of a polynomial intersects the x-axis at only one point, it cannot be a quadratic polynomial" is true or false, let's analyze it step by step.

### Step 1: Understanding the Graph of a Quadratic Polynomial
A quadratic polynomial can be expressed in the standard form as:
\[ f(x) = ax^2 + bx + c \]
where \( a \neq 0 \). The graph of a quadratic polynomial is a parabola.

### Step 2: Conditions for Intersection with the X-axis
The points where the graph intersects the x-axis are the roots of the polynomial. A quadratic polynomial can have:
- Two distinct real roots (intersects the x-axis at two points)
- One real root (intersects the x-axis at one point, also known as a repeated root)
- No real roots (does not intersect the x-axis)

### Step 3: Analyzing the Case of One Intersection Point
When a quadratic polynomial intersects the x-axis at only one point, it means that it has a repeated root. This occurs when the discriminant (\( D \)) of the quadratic equation is zero:
\[ D = b^2 - 4ac = 0 \]
In this case, the graph touches the x-axis at one point (the vertex of the parabola) but does not cross it.

### Step 4: Example of a Quadratic Polynomial with One Intersection
Consider the quadratic polynomial:
\[ f(x) = (x - 2)^2 \]
This polynomial has its vertex at \( x = 2 \) and touches the x-axis at the point \( (2, 0) \). Here, it intersects the x-axis at only one point, yet it is still a quadratic polynomial.

### Conclusion
Since we have shown that a quadratic polynomial can intersect the x-axis at only one point (with a repeated root), the statement "If the graph of a polynomial intersects the x-axis at only one point, it cannot be a quadratic polynomial" is **False**.

### Justification
The statement is false because a quadratic polynomial can indeed intersect the x-axis at only one point if it has a repeated root, as demonstrated with the example \( f(x) = (x - 2)^2 \).

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