If the graph of a polynomial crosses the x-axis at the origin (0,0), which conclusion is certain?

(0) is a zero of the polynomial

If the graph passes through (0,0) then by definition the polynomial satisfies \(p(0)=0\), so 0 is a root. This does not imply the polynomial is identically zero (option C is false) — a single zero at x=0 does not make the function zero everywhere. Option B contradicts the given point because a nonzero constant term would give \(p(0)\neq0\). Option D is wrong because "touching" (tangent) the x-axis indicates a repeated (even multiplicity) root, whereas "crossing" indicates the graph passes through the axis; from the fact it crosses we only can be certain that \(p(0)=0\). Exam tip: to check whether \(a\) is a root, substitute \(x=a\) into \(p(x)\); if \(p(a)=0\), then \(a\) is a zero of the polynomial.

This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Geometrical meaning of the zeroes of a polynomial..