If the graph of a polynomial cuts the \(x\)-axis at \(a\) and \(b\) with \(a \neq b\), what are the values of \(p(a)\) and \(p(b)\)?
Answer and explanation
Correct answer: Both are \(0\)
If the graph of a polynomial meets the \(x\)-axis at a point, the polynomial's value at that x-coordinate is zero. Hence when the graph crosses the axis at \(a\) and \(b\), we have \(p(a)=0\) and \(p(b)=0\). Even if a root has multiplicity greater than one (the graph just touches the axis), the value at that root is still zero. Option D (“they are not equal”) is incorrect because both values are equal (both zero); option C is also incorrect because the axis-intercept cannot produce 0 at one point and 1 at the other. Exam tip: whenever asked about x-intercepts, immediately set the polynomial value to 0 at those x-values — that gives the roots directly.
Frequently asked questions
What is the correct answer to this question?
Both are \(0\)
Why is this the correct answer?
If the graph of a polynomial meets the \(x\)-axis at a point, the polynomial's value at that x-coordinate is zero. Hence when the graph crosses the axis at \(a\) and \(b\), we have \(p(a)=0\) and \(p(b)=0\). Even if a root has multiplicity greater than one (the graph just touches the axis), the value at that root is still zero. Option D (“they are not equal”) is incorrect because both values are equal (both zero); option C is also incorrect because the axis-intercept cannot produce 0 at one point and 1 at the other. Exam tip: whenever asked about x-intercepts, immediately set the polynomial value to 0 at those x-values — that gives the roots directly.
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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