A quadratic polynomial graph does not touch or intersect the x-axis. What is the number of distinct real zeroes?
Answer and explanation
Correct answer: Zero
A real zero of a polynomial is a real number r for which p(r) = 0. On the graph y = p(x), this means that the graph must have a point (r, 0), so it must intersect or touch the x-axis. The statement says that the quadratic graph neither touches nor intersects the x-axis. Consequently, there is no real x-coordinate at which p(x) becomes zero, and the number of distinct real zeroes is zero. In discriminant language, a quadratic with no real x-axis contact has discriminant less than zero, although calculating the discriminant is not necessary here. One or two would require contact with the axis; infinitely many is impossible for a nonzero quadratic.
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What is the correct answer to this question?
Zero
Why is this the correct answer?
A real zero of a polynomial is a real number r for which p(r) = 0. On the graph y = p(x), this means that the graph must have a point (r, 0), so it must intersect or touch the x-axis. The statement says that the quadratic graph neither touches nor intersects the x-axis. Consequently, there is no real x-coordinate at which p(x) becomes zero, and the number of distinct real zeroes is zero. In discriminant language, a quadratic with no real x-axis contact has discriminant less than zero, although calculating the discriminant is not necessary here. One or two would require contact with the axis; infinitely many is impossible for a nonzero quadratic.
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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