Question:
Observe the graph of polynomial p(x). Number of zeroes of p(x) is
Observe the graph of polynomial p(x). Number of zeroes of p(x) is
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Do not count the intersection with the vertical \(y\)-axis.
Only the points on the horizontal \(x\)-axis represent values of \(x\) where the polynomial evaluates to zero, i.e., \(p(x) = 0\).
Only the points on the horizontal \(x\)-axis represent values of \(x\) where the polynomial evaluates to zero, i.e., \(p(x) = 0\).
Updated On: Jul 9, 2026
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
The topic of this question is Polynomials, specifically focusing on the graphical representation of their zeroes.
The zeroes of a polynomial \(p(x)\) are the values of \(x\) for which \(p(x) = 0\).
On a Cartesian coordinate grid, these correspond to the \(x\)-coordinates of the points where the graph of the equation \(y = p(x)\) intersects or touches the horizontal \(x\)-axis.
We are asked to find the number of zeroes of \(p(x)\) by observing the provided graph.
Step 2: Key Formula or Approach:
To find the number of zeroes of \(p(x)\) from its graph:
- Trace the curve and identify all points where it crosses or touches the horizontal \(x\)-axis.
- The total count of these distinct intersection points equals the number of real zeroes.
- Intersections with the vertical \(y\)-axis are not counted as zeroes of \(p(x)\).
Step 3: Detailed Explanation:
• Trace the given curve \(y = p(x)\) from left to right along the horizontal axis:
• Point 1: The curve crosses the negative \(x\)-axis at one point on the left side of the origin. This is the first zero.
• Point 2: The curve crosses the \(x\)-axis at the origin \(O(0, 0)\). This is the second zero.
• Point 3: The curve goes below the \(x\)-axis, turns upwards, and crosses the positive \(x\)-axis at another point. This is the third zero.
• Count the total number of intersection points with the \(x\)-axis:
The curve intersects the horizontal \(x\)-axis at exactly 3 distinct points.
Therefore, the polynomial has exactly 3 real zeroes.
Step 4: Final Answer:
The number of zeroes of the polynomial \(p(x)\) is 3.
Therefore, the correct option is (D).
The topic of this question is Polynomials, specifically focusing on the graphical representation of their zeroes.
The zeroes of a polynomial \(p(x)\) are the values of \(x\) for which \(p(x) = 0\).
On a Cartesian coordinate grid, these correspond to the \(x\)-coordinates of the points where the graph of the equation \(y = p(x)\) intersects or touches the horizontal \(x\)-axis.
We are asked to find the number of zeroes of \(p(x)\) by observing the provided graph.
Step 2: Key Formula or Approach:
To find the number of zeroes of \(p(x)\) from its graph:
- Trace the curve and identify all points where it crosses or touches the horizontal \(x\)-axis.
- The total count of these distinct intersection points equals the number of real zeroes.
- Intersections with the vertical \(y\)-axis are not counted as zeroes of \(p(x)\).
Step 3: Detailed Explanation:
• Trace the given curve \(y = p(x)\) from left to right along the horizontal axis:
• Point 1: The curve crosses the negative \(x\)-axis at one point on the left side of the origin. This is the first zero.
• Point 2: The curve crosses the \(x\)-axis at the origin \(O(0, 0)\). This is the second zero.
• Point 3: The curve goes below the \(x\)-axis, turns upwards, and crosses the positive \(x\)-axis at another point. This is the third zero.
• Count the total number of intersection points with the \(x\)-axis:
The curve intersects the horizontal \(x\)-axis at exactly 3 distinct points.
Therefore, the polynomial has exactly 3 real zeroes.
Step 4: Final Answer:
The number of zeroes of the polynomial \(p(x)\) is 3.
Therefore, the correct option is (D).
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