If the graph of p(x) crosses the x-axis at (2, 0) and touches it at (-1, 0), how many distinct real zeros does p(x) have?
Answer and explanation
Correct answer: Two
Crossing the x-axis at a point indicates a root with odd multiplicity (a simple real zero); crossing at (2,0) gives the root x=2. Touching the axis indicates a root with even multiplicity; touching at (−1,0) gives the root x=−1. These are two distinct x-values, so there are 2 distinct real zeros. Why "three" is wrong: touching does not create an extra distinct x-value, it only implies even multiplicity at x=−1. Exam tip: count distinct x-values of intercepts; use multiplicity only to determine crossing vs touching behavior.
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What is the correct answer to this question?
Two
Why is this the correct answer?
Crossing the x-axis at a point indicates a root with odd multiplicity (a simple real zero); crossing at (2,0) gives the root x=2. Touching the axis indicates a root with even multiplicity; touching at (−1,0) gives the root x=−1. These are two distinct x-values, so there are 2 distinct real zeros. Why "three" is wrong: touching does not create an extra distinct x-value, it only implies even multiplicity at x=−1. Exam tip: count distinct x-values of intercepts; use multiplicity only to determine crossing vs touching behavior.
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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