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
Show Hint
Be careful not to include the point of intersection on the vertical \(y\)-axis when counting the zeroes of a polynomial in \(x\).
Zeroes are only found where the curve intersects or touches the horizontal \(x\)-axis (where \(y = 0\)).
Zeroes are only found where the curve intersects or touches the horizontal \(x\)-axis (where \(y = 0\)).
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 interpretation of their zeroes.
The zeroes of a polynomial \(p(x)\) are the values of \(x\) for which \(p(x) = 0\).
When we plot the graph of \(y = p(x)\) on a Cartesian coordinate system, the zeroes correspond to the points where the curve intersects or touches the horizontal \(x\)-axis.
We are asked to find the number of zeroes of the polynomial \(p(x)\) by observing its given graph.
Step 2: Key Formula or Approach:
To find the number of zeroes of the polynomial \(p(x)\) from its graph:
- Look at the horizontal \(x\)-axis.
- Count the total number of distinct points where the graph intersects (crosses) or touches the \(x\)-axis.
- Ignore any intersection points with the vertical \(y\)-axis, as they represent \(p(0)\), which is not a zero of the polynomial.
Step 3: Detailed Explanation:
• Let us analyze the given graph of \(y = p(x)\) by tracing the curve from left to right along the \(x\)-axis:
• Point 1: The curve crosses the negative \(x\)-axis at one point on the left side of the origin. This represents the first real zero of the polynomial.
• Point 2: The curve crosses the positive \(x\)-axis at a point near the origin. This represents the second real zero of the polynomial.
• Point 3: After going below the \(x\)-axis and crossing the negative \(y\)-axis, the curve turns upwards and crosses the positive \(x\)-axis once more. This represents the third real zero of the polynomial.
• Count the total number of intersections with the horizontal \(x\)-axis:
The curve intersects the \(x\)-axis at exactly 3 distinct points.
Therefore, the polynomial \(p(x)\) 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 interpretation of their zeroes.
The zeroes of a polynomial \(p(x)\) are the values of \(x\) for which \(p(x) = 0\).
When we plot the graph of \(y = p(x)\) on a Cartesian coordinate system, the zeroes correspond to the points where the curve intersects or touches the horizontal \(x\)-axis.
We are asked to find the number of zeroes of the polynomial \(p(x)\) by observing its given graph.
Step 2: Key Formula or Approach:
To find the number of zeroes of the polynomial \(p(x)\) from its graph:
- Look at the horizontal \(x\)-axis.
- Count the total number of distinct points where the graph intersects (crosses) or touches the \(x\)-axis.
- Ignore any intersection points with the vertical \(y\)-axis, as they represent \(p(0)\), which is not a zero of the polynomial.
Step 3: Detailed Explanation:
• Let us analyze the given graph of \(y = p(x)\) by tracing the curve from left to right along the \(x\)-axis:
• Point 1: The curve crosses the negative \(x\)-axis at one point on the left side of the origin. This represents the first real zero of the polynomial.
• Point 2: The curve crosses the positive \(x\)-axis at a point near the origin. This represents the second real zero of the polynomial.
• Point 3: After going below the \(x\)-axis and crossing the negative \(y\)-axis, the curve turns upwards and crosses the positive \(x\)-axis once more. This represents the third real zero of the polynomial.
• Count the total number of intersections with the horizontal \(x\)-axis:
The curve intersects the \(x\)-axis at exactly 3 distinct points.
Therefore, the polynomial \(p(x)\) 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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