If p(2) is positive and p(5) is negative, which statement about the graph is most appropriate?
Answer and explanation
Correct answer: The graph may cross the x-axis between x = 2 and x = 5
Polynomial functions are continuous: their graphs have no jumps or breaks. Since p(2) is positive, the graph is above the x-axis at x = 2; since p(5) is negative, it is below the x-axis at x = 5. Moving continuously from x = 2 to x = 5, the graph must pass through y = 0 at least once. Thus it has at least one real zero in the interval (2, 5), so it may cross the x-axis between those values. The wording “may cross” is appropriate because the information does not determine the exact zero or exclude multiple crossings. Options B and C are unsupported, and a polynomial graph cannot be a vertical line parallel to the y-axis.
Frequently asked questions
What is the correct answer to this question?
The graph may cross the x-axis between x = 2 and x = 5
Why is this the correct answer?
Polynomial functions are continuous: their graphs have no jumps or breaks. Since p(2) is positive, the graph is above the x-axis at x = 2; since p(5) is negative, it is below the x-axis at x = 5. Moving continuously from x = 2 to x = 5, the graph must pass through y = 0 at least once. Thus it has at least one real zero in the interval (2, 5), so it may cross the x-axis between those values. The wording “may cross” is appropriate because the information does not determine the exact zero or exclude multiple crossings. Options B and C are unsupported, and a polynomial graph cannot be a vertical line parallel to the y-axis.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Geometrical meaning of the zeroes of a polynomial..
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