Assertion (A) If the graph of a polynomial intersects the x-axis at exactly two points, then the number of zeroes of that polynomial is 2. Reason (R) The number of zeroes of a polynomial is equal to the number of points where the graph of the polynomial intersects x-axis.

Step 1: Analyze Assertion (A)

The zeroes of a polynomial are the values of $x$ for which the polynomial's value is zero. Graphically, these are the points where the graph of the polynomial crosses or touches the $x$ -axis. If the graph intersects the $x$ -axis at exactly two distinct points, it means there are two distinct values of $x$ for which the polynomial is zero. Therefore, the number of zeroes is indeed 2.

Step 2: Analyze Reason (R)

This statement defines the fundamental relationship between the zeroes of a polynomial and its graph. Each point where the graph of a polynomial intersects the $x$ -axis corresponds to a real root or zero of the polynomial. Conversely, every real zero of a polynomial corresponds to an intersection point on the $x$ -axis. Thus, the number of real zeroes is precisely the number of $x$ -intercepts.

Step 3: Evaluate Relationship between A and R

Both the assertion and the reason are true statements. Furthermore, the reason directly explains why the assertion is true. The definition provided in Reason (R) is the basis for understanding why two intersection points on the $x$ -axis imply two zeroes for the polynomial, as stated in Assertion (A).

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