A polynomial's graph crosses the \(x\)-axis at \\((-5,0)\\) and touches the \(x\)-axis at \\((2,0)\\). Based on this information, what is the number of real (distinct) zeros?
Answer and explanation
Correct answer: Two
Crossing at \((-5,0)\) means \(x=-5\) is a real root (typically odd multiplicity). Touching at \((2,0)\) means \(x=2\) is also a real root (typically even multiplicity, e.g. multiplicity 2). These are two distinct real zeros, so the answer is two. The option "three" might arise if one counts multiplicities (for example multiplicity 1 at \(-5\) and multiplicity 2 at \(2\) gives three roots counting multiplicity), but the question asks for distinct real zeros. Exam tip: always check whether the question asks for distinct roots or roots counted with multiplicity when interpreting touch vs. cross.
Frequently asked questions
What is the correct answer to this question?
Two
Why is this the correct answer?
Crossing at \((-5,0)\) means \(x=-5\) is a real root (typically odd multiplicity). Touching at \((2,0)\) means \(x=2\) is also a real root (typically even multiplicity, e.g. multiplicity 2). These are two distinct real zeros, so the answer is two. The option "three" might arise if one counts multiplicities (for example multiplicity 1 at \(-5\) and multiplicity 2 at \(2\) gives three roots counting multiplicity), but the question asks for distinct real zeros. Exam tip: always check whether the question asks for distinct roots or roots counted with multiplicity when interpreting touch vs. cross.
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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