If p(x)=x^3-16x, at how many distinct points will the graph cut the x-axis?
Answer and explanation
Correct answer: Three
The governing concept is that distinct real zeroes correspond to distinct points where the graph meets the x-axis. First factor out the common factor x: x^3-16x = x(x^2-16). Then use the difference of squares: x(x^2-16)=x(x-4)(x+4). Setting each factor equal to zero gives x=0, x=4 and x=-4. These are three different real x-values, so the graph has three distinct x-axis intersection points. Option C is correct. Option A omits two roots, Option B omits one root, and Option D incorrectly counts beyond the three factors or treats multiplicity as an additional distinct point.
Frequently asked questions
What is the correct answer to this question?
Three
Why is this the correct answer?
The governing concept is that distinct real zeroes correspond to distinct points where the graph meets the x-axis. First factor out the common factor x: x^3-16x = x(x^2-16). Then use the difference of squares: x(x^2-16)=x(x-4)(x+4). Setting each factor equal to zero gives x=0, x=4 and x=-4. These are three different real x-values, so the graph has three distinct x-axis intersection points. Option C is correct. Option A omits two roots, Option B omits one root, and Option D incorrectly counts beyond the three factors or treats multiplicity as an additional distinct point.
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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