If (p(x)=x^3-25x), at how many distinct points will the graph cut the (x)-axis?
Answer and explanation
Correct answer: Three
To find where the graph meets the x-axis, solve \(p(x)=0\). For the given polynomial, \(x^3-25x=0\). Taking the common factor x gives \(x(x^2-25)=0\), and the difference of squares factors the second part as \(x(x-5)(x+5)=0\). A product is zero when at least one factor is zero, so the possible x-values are \(0\), 5, and -5.
These three values are distinct, so the graph has three different x-axis intersection points: \((0,0)\), \((5,0)\), and \((-5,0)\). Repeated factors, if present, would affect multiplicity but not create an additional distinct point. Since the question asks for distinct points, the answer is three, which is option C.
Frequently asked questions
What is the correct answer to this question?
Three
Why is this the correct answer?
To find where the graph meets the x-axis, solve \(p(x)=0\). For the given polynomial, \(x^3-25x=0\). Taking the common factor x gives \(x(x^2-25)=0\), and the difference of squares factors the second part as \(x(x-5)(x+5)=0\). A product is zero when at least one factor is zero, so the possible x-values are \(0\), 5, and -5.
These three values are distinct, so the graph has three different x-axis intersection points: \((0,0)\), \((5,0)\), and \((-5,0)\). Repeated factors, if present, would affect multiplicity but not create an additional distinct point. Since the question asks for distinct points, the answer is three, which is option C.
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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