Given: The graphs of $y = p(x)$ for various polynomials $p(x)$ are provided in the figure.

To Find: The number of zeroes of the polynomial $p(x)$ in each of the six given cases.

Theoretical Basis:

The zeroes of a polynomial $p(x)$ are the $x$-coordinates of the points where the graph of $y = p(x)$ intersects the $x$-axis. Geometrically, the number of zeroes is equal to the number of times the graph of the polynomial intersects or touches the $x$-axis.

Visual Representation (Conceptual Diagram):

Step-by-Step Analysis:

Case (i): The graph is a straight line parallel to the $x$-axis. It does not intersect the $x$-axis at any point.
[Since there are no points of intersection with the $x$-axis]
Number of zeroes = 0

Case (ii): The graph intersects the $x$-axis at exactly one point.
[Since the graph crosses the $x$-axis once]
Number of zeroes = 1

Case (iii): The graph intersects the $x$-axis at three distinct points.
[Since the graph crosses the $x$-axis at three separate locations]
Number of zeroes = 3

Case (iv): The graph is a parabola opening upwards, intersecting the $x$-axis at two distinct points.
[Since the graph crosses the $x$-axis at two locations]
Number of zeroes = 2

Case (v): The graph intersects the $x$-axis at four distinct points.
[Since the graph crosses the $x$-axis at four locations]
Number of zeroes = 4

Case (vi): The graph intersects the $x$-axis at one point and touches it at two other points. In total, there are three distinct points of contact/intersection.
[Since the graph touches or crosses the $x$-axis at three locations]
Number of zeroes = 3

Final Answer:

The number of zeroes for each case are: (i) 0, (ii) 1, (iii) 3, (iv) 2, (v) 4, and (vi) 3.