Question:
The graph of a polynomial \(p(x)\) is shown here. The number of zeroes of the polynomial \(p(x)\) is
The graph of a polynomial \(p(x)\) is shown here. The number of zeroes of the polynomial \(p(x)\) is
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Be careful not to count intersections with the vertical \(y\)-axis. Only count points where the curve crosses or touches the horizontal \(x\)-axis.
Updated On: Jun 25, 2026
- 5
- 1
- 0
- 4
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
The question asks us to find the number of zeroes of the polynomial \(p(x)\) from its given graph.
A zero of a polynomial \(p(x)\) corresponds to a value of \(x\) for which \(p(x) = 0\).
Step 2: Key Formula or Approach:
In a coordinate plane graph of \(y = p(x)\), the zeroes of the polynomial are the \(x\)-coordinates of the points where the graph intersects or touches the \(x\)-axis.
Thus: \[ \text{Number of zeroes} = \text{Number of intersection points with the } x\text{-axis} \]
Step 3: Detailed Explanation:
1. Look closely at the provided graph of the curve \(y = p(x)\).
2. Trace the curve along the horizontal \(x\)-axis to identify points of intersection: - The curve crosses the \(x\)-axis at one point on the negative side (far left).
- The curve crosses the \(x\)-axis a second time closer to the origin on the negative side.
- The curve crosses the \(x\)-axis a third time on the positive side.
- The curve crosses the \(x\)-axis a fourth time on the positive side (far right).
3. Counting these points, we find that the graph intersects the horizontal axis at exactly 4 distinct points.
4. Since there are 4 intersection points, the polynomial \(p(x)\) has exactly 4 real zeroes.
Step 4: Final Answer:
The number of zeroes of the given polynomial \(p(x)\) is 4.
Hence, the correct option is (D).
The question asks us to find the number of zeroes of the polynomial \(p(x)\) from its given graph.
A zero of a polynomial \(p(x)\) corresponds to a value of \(x\) for which \(p(x) = 0\).
Step 2: Key Formula or Approach:
In a coordinate plane graph of \(y = p(x)\), the zeroes of the polynomial are the \(x\)-coordinates of the points where the graph intersects or touches the \(x\)-axis.
Thus: \[ \text{Number of zeroes} = \text{Number of intersection points with the } x\text{-axis} \]
Step 3: Detailed Explanation:
1. Look closely at the provided graph of the curve \(y = p(x)\).
2. Trace the curve along the horizontal \(x\)-axis to identify points of intersection: - The curve crosses the \(x\)-axis at one point on the negative side (far left).
- The curve crosses the \(x\)-axis a second time closer to the origin on the negative side.
- The curve crosses the \(x\)-axis a third time on the positive side.
- The curve crosses the \(x\)-axis a fourth time on the positive side (far right).
3. Counting these points, we find that the graph intersects the horizontal axis at exactly 4 distinct points.
4. Since there are 4 intersection points, the polynomial \(p(x)\) has exactly 4 real zeroes.
Step 4: Final Answer:
The number of zeroes of the given polynomial \(p(x)\) is 4.
Hence, the correct option is (D).
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