If no point of a polynomial graph lies on the x-axis, which statement is correct?
Answer and explanation
Correct answer: It has no real zero
For a polynomial p(x), a real zero is a real number r satisfying p(r) = 0. On the coordinate graph y = p(x), this condition produces the point (r, 0), which lies on the x-axis. Thus real zeroes are represented exactly by x-axis intersections. If no point of the graph lies on that axis, there is no real value of x for which p(x) = 0. Therefore the polynomial has no real zero, as stated in Option A. Options B and C claim intersections that the question explicitly rules out. Option D is also false because a polynomial cannot have every number as a zero unless it is the zero polynomial, whose graph is the x-axis itself.
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What is the correct answer to this question?
It has no real zero
Why is this the correct answer?
For a polynomial p(x), a real zero is a real number r satisfying p(r) = 0. On the coordinate graph y = p(x), this condition produces the point (r, 0), which lies on the x-axis. Thus real zeroes are represented exactly by x-axis intersections. If no point of the graph lies on that axis, there is no real value of x for which p(x) = 0. Therefore the polynomial has no real zero, as stated in Option A. Options B and C claim intersections that the question explicitly rules out. Option D is also false because a polynomial cannot have every number as a zero unless it is the zero polynomial, whose graph is the x-axis itself.
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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