If the graph of a polynomial crosses the x-axis at \(x=1\) and only touches the x-axis at \(x=6\), how many distinct real zeros does the polynomial have?
Answer and explanation
Correct answer: Two
A crossing at a point means the zero has odd multiplicity and a touch means even multiplicity, but in both cases the polynomial satisfies \(p(x)=0\). The x‑values given are \(x=1\) and \(x=6\), so there are two distinct real zeros. The distractor “three” is incorrect because multiplicities (odd/even) do not create additional distinct x‑values. Exam tip: count distinct x‑values where the graph meets the axis; use crossing vs touching only to infer multiplicity, not the number of distinct zeros.
Frequently asked questions
What is the correct answer to this question?
Two
Why is this the correct answer?
A crossing at a point means the zero has odd multiplicity and a touch means even multiplicity, but in both cases the polynomial satisfies \(p(x)=0\). The x‑values given are \(x=1\) and \(x=6\), so there are two distinct real zeros. The distractor “three” is incorrect because multiplicities (odd/even) do not create additional distinct x‑values. Exam tip: count distinct x‑values where the graph meets the axis; use crossing vs touching only to infer multiplicity, not the number of distinct zeros.
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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