Look at the graphs in Fig. 2.9 given below. Each is the graph of $y = p(x)$, where $p(x)$ is a polynomial. For each of the graphs, find the number of zeroes of $p(x)$.

Step 1: Understand the Definition of Zeroes

A zero of a polynomial $p(x)$ is a value of $x$ for which $p(x) = 0$. Geometrically, this corresponds to the points where the graph of $y = p(x)$ crosses or touches the $x$ -axis. We need to count these intersection points for each given graph.

Step 2: Analyze Graph (i)

Observe the first graph. The curve intersects the $x$ -axis at exactly one point. Therefore, the polynomial $p(x)$ represented by this graph has one zero. As per NCERT Class 10 Mathematics, page 1, Example 1 (i) states 'The number of zeroes is 1 as the graph intersects the x-axis at one point only'.

Step 3: Analyze Graph (ii)

Look at the second graph. The parabolic curve intersects the $x$ -axis at two distinct points. This means the polynomial $p(x)$ has two zeroes. As per NCERT Class 10 Mathematics, page 1, Example 1 (ii) states 'The number of zeroes is 2 as the graph intersects the x-axis at two points'.

Step 4: Analyze Graph (iii)

Examine the third graph. The curve intersects the $x$ -axis at three distinct points. Hence, the polynomial $p(x)$ has three zeroes. For example, a cubic polynomial like $y = x^3 - x^2$ (NCERT Class 10 Mathematics, page 1, Fig. 2.8) can intersect the $x$ -axis at three points.

Step 5: Analyze Graph (iv)

Consider the fourth graph. The curve intersects the $x$ -axis at exactly one point. Therefore, the polynomial $p(x)$ has one zero. A cubic polynomial can have one zero, as seen in NCERT Class 10 Mathematics, page 1, Fig. 2.7 for $y=x^3$.

Step 6: Analyze Graph (v)

Observe the fifth graph. The curve touches the $x$ -axis at exactly one point. Even though it touches rather than crosses, it still counts as one intersection point, specifically a repeated root (zero of multiplicity greater than 1). Thus, the polynomial $p(x)$ has one zero.

Step 7: Analyze Graph (vi)

Finally, look at the sixth graph. The curve intersects the $x$ -axis at four distinct points. This indicates that the polynomial $p(x)$ has four zeroes. A polynomial of degree $n$ can have at most $n$ zeroes (NCERT Class 10 Mathematics, page 1).

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