If the graph of a polynomial \(p(x)\) intersects the x-axis at the point \((r,0)\), which of the following statements is always true?
Answer and explanation
Correct answer: \(p(r)=0\)
The point \((r,0)\) has x‑coordinate \(r\) and y‑coordinate 0. For the polynomial \(p(x)\), the y‑value at x=\(r\) is \(p(r)\); therefore \(p(r)=0\). Option B (\(p(0)=r\)) incorrectly relates the value at x=0 to r and need not hold. Option C (\(r=0\)) is only true if the intercept happens at the origin, not in general. Option D (\(p(r)=r\)) would mean the function value equals the x‑coordinate at that point, which contradicts y=0 here. Exam tip: Whenever you see an x‑intercept \((a,0)\), immediately record \(p(a)=0\) — that a is a root of the polynomial.
Frequently asked questions
What is the correct answer to this question?
\(p(r)=0\)
Why is this the correct answer?
The point \((r,0)\) has x‑coordinate \(r\) and y‑coordinate 0. For the polynomial \(p(x)\), the y‑value at x=\(r\) is \(p(r)\); therefore \(p(r)=0\). Option B (\(p(0)=r\)) incorrectly relates the value at x=0 to r and need not hold. Option C (\(r=0\)) is only true if the intercept happens at the origin, not in general. Option D (\(p(r)=r\)) would mean the function value equals the x‑coordinate at that point, which contradicts y=0 here. Exam tip: Whenever you see an x‑intercept \((a,0)\), immediately record \(p(a)=0\) — that a is a root of the polynomial.
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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