If the graph of a polynomial crosses the x-axis at \(x=-2\) and only touches it at \(x=3\), what are the distinct real zeros?
Answer and explanation
Correct answer: \(-2\) and \(3\)
Both crossing and touching correspond to roots: crossing at \(x=-2\) indicates an odd-multiplicity root (the sign of \(p(x)\) changes), while touching at \(x=3\) indicates an even-multiplicity root (no sign change). Therefore the distinct real zeros are \(x=-2\) and \(x=3\). The closest distractor 'Only -2' is incorrect because touching at 3 still gives \(p(3)=0\). Exam tip: list distinct zeros as the x-values (comma-separated) and note multiplicities if required.
Frequently asked questions
What is the correct answer to this question?
\(-2\) and \(3\)
Why is this the correct answer?
Both crossing and touching correspond to roots: crossing at \(x=-2\) indicates an odd-multiplicity root (the sign of \(p(x)\) changes), while touching at \(x=3\) indicates an even-multiplicity root (no sign change). Therefore the distinct real zeros are \(x=-2\) and \(x=3\). The closest distractor 'Only -2' is incorrect because touching at 3 still gives \(p(3)=0\). Exam tip: list distinct zeros as the x-values (comma-separated) and note multiplicities if required.
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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