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).
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Step-by-Step Solution
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=x3−x2 (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=x3.
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).