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). Fig. 2.9
Step-by-Step Solution
Key Concept: A zero (root) of a polynomial p(x) is a value of x for which p(x)=0. Graphically, this corresponds to the points where the curve y = p(x) intersects the x‑axis. Hence, the number of zeroes equals the number of distinct x‑intercepts of the graph.
1. Identify the x‑axis – In each sub‑figure (a), (b), (c) and (d) draw an imaginary horizontal line representing the x‑axis.
2. Count the intersections – Observe how many times the curve meets this line.
- Figure (a): The curve cuts the x‑axis at two distinct points. Therefore, p(x) has 2 zeroes.
- Figure (b): The curve touches the x‑axis at exactly one point (tangent) and does not cross it elsewhere. Hence, p(x) has 1 zero.
- Figure (c): The curve crosses the x‑axis three times, giving three distinct points where p(x)=0. Thus, p(x) has 3 zeroes.
- Figure (d): The curve never meets the x‑axis; it stays entirely above (or below) it. Consequently, p(x) has 0 zeroes.
3. Write the answers – Summarize the counts for each graph.
- (a) 2 zeroes
- (b) 1 zero
- (c) 3 zeroes
- (d) 0 zeroes
Correct Answer: (a) 2, (b) 1, (c) 3, (d) 0