Geometrical meaning of the zeroes of a polynomial.

For a function or polynomial the graph contains points of the form \\((a,p(a))\\). If \(p(0)=0\), then at \(x=0\) we have \(y=p(0)=0\), so the graph passes through \\((0,0)\\) — the origin. Option (1,0) is a plausible distractor but is wrong because \(p(1)\) is not given; you cannot assume \(p(1)=0\). Quick exam tip: evaluate the polynomial at the given x; whenever \(p(a)=0\), the graph crosses the x-axis at \\((a,0)\\).

A polynomial intersects the x-axis at a point only if its value (y) at that x is zero. To intersect at x = 0 we must have p(0) = 0. Option B gives p(0)=5 so the value is not zero and the graph does not meet the x-axis at x=0. Option C has a zero at x=1, so the intersection is at x=1, not x=0. Option D yields p(-1)=1, which is not zero. Exam tip: substitute the specified x into the polynomial — if the result is 0, the graph crosses the x-axis at that x-coordinate.

The zero of a polynomial is the x-value where the polynomial equals zero, i.e., where the graph meets the x-axis (y=0). At the point \((a,0)\) the x-coordinate is \(a\), so the zero is \(a\). Option \(0\) is incorrect because that is the y-coordinate at the intersection; \(-a\) and \(a^2\) are not the x-coordinate of the given point. Exam tip: when a graph intersects the x-axis, read off the x-coordinate — that is the zero (root).

Zeroes are the (x)-values where the graph cuts or touches the (x)-axis. Hence they are linked with (x)-intercepts.

Both cutting and touching count as meeting the (x)-axis. If the two points are distinct, there are two distinct real zeroes.

The point \\( (8,0) \\) means that at x=8 the function value y is 0. Hence \\(p(8)=0\\). Option \\(8\\) is a common trap — it is the x-coordinate, not the value of the polynomial. Exam tip: any intersection with the x-axis gives a root, so substitute the x-value to get zero.

\(p(-6)=0\) means the polynomial equals zero at \(x=-6\), so the graph has \(y=0\) when \(x=-6\). Hence the point \((-6,0)\) lies on the graph. Option B is wrong because it uses \(x=6\) instead of \(x=-6\). Options C and D are y‑intercepts (x=0) and do not represent a zero of the polynomial. Exam tip: a root/zero \(p(a)=0\) corresponds to the x‑intercept \((a,0)\).

Real zeroes are obtained from common points of the graph and the (x)-axis. If there is no common point, there is no real zero.

A point where the graph touches the x-axis is a real root because at that point p(x)=0. "Touches" usually indicates an even multiplicity root, but regardless of multiplicity it is still a zero of the polynomial. Thus the distinct real zero is x = -2. Option B (2) is wrong because the given x-coordinate is -2, not +2; option C (0) is wrong because the origin is not the given intercept; option D (None) is wrong because a touching point is indeed a root. Exam tip: remember that "touches" implies an even multiplicity root, while "cuts" implies an odd multiplicity.

The governing concept is the geometrical meaning of a zero of a polynomial. A number x = a is a zero when the value of the polynomial is zero, that is, f(a) = 0. On a graph, f(x) is represented by the y-coordinate, so f(x) becomes zero exactly where the graph lies on the x-axis. The points given are (r, 0) and (s, 0). Their x-coordinates are r and s, and the common y-coordinate 0 shows that the polynomial value is zero at both inputs. Therefore the zeroes are r and s, so option A is correct. The coordinates 0, 0 are not the zeroes; the negative values are unjustified, and r + s and rs are relationships involving the zeroes, not the zeroes themselves.

For a number \(x=a\) to be a root of a polynomial we must have \(P(a)=0\). Thus the value of the polynomial at that x must be zero, so option A is correct. The constant term (option D) may be zero in some polynomials but it is not a necessary condition (e.g. \(P(x)=x-1\) has constant term \(-1\) yet root at \(x=1\)). Coefficients or exponents being zero do not by themselves indicate a root. Exam tip: on the graph, roots are the x-intercepts where \(y=0\); read off x-coordinates of those intercepts.

Zeroes are the (x)-coordinates of the points where the graph cuts the (x)-axis. Hence the zeroes are (-4) and (1).

A zero of a polynomial is an x-value for which the polynomial has value 0; on its graph, this is a point on the x-axis. The given point is (6,0), so the polynomial has value 0 at x = 6. Therefore, its distinct real zero is 6. The option -6 is incorrect because the graph does not touch the x-axis at x = -6, and 0 would require the point (0,0). Exam tip: From an x-axis point (a,0), directly identify a as a zero of the polynomial.

\(p(9)=0\) means the polynomial takes value zero when \(x=9\), so \(y=0\) at that x. Hence the graph meets the x-axis at \((9,0)\). Why others are wrong: \((-9,0)\) has the wrong sign for x, \((0,9)\) swaps x and y, and \((9,9)\) does not have y=0. Exam tip: a root \(p(a)=0\) always corresponds to the x-intercept \((a,0)\).

A zero (root) of a polynomial is an x-value for which \(p(x)=0\). Since \(p(-3)=0\) is given, \(-3\) is a root of the polynomial. The closest distractor, 3, is incorrect because the condition is satisfied only at \(-3\) as stated; there is no information that \(p(3)=0\). Exam tip: To identify a root, substitute the candidate value into \(p(x)\) — if the result is zero, it is a root.

Zeroes are linked to the (x)-axis where (y=0). A (y)-axis intersection only shows the value of the polynomial at (x=0).

Each distinct intersection with the (x)-axis gives one real zero. Therefore three distinct intersections give three real zeroes.

Real zeroes are found only when the graph meets the (x)-axis. With no common point, there are (0) real zeroes.

The zeroes of a polynomial are the x-coordinates where its graph meets the x-axis. A graph that crosses the axis usually gives a zero with a sign change, while a graph that only touches the axis and turns back gives a repeated zero. Although the zero may be repeated algebraically, its distinct real value is counted as one real zero in this question.

The parabola touches the x-axis at the single point \((3,0)\). Thus the corresponding value of x is \(x=3\), and there is no second point of intersection. For example, a polynomial such as \((x-3)^2\) has the repeated root 3, but only one distinct real zero. Therefore option A, one, is correct.

The zeroes (roots) of a polynomial/ parabola are the x-coordinates where the graph meets the x-axis (y = 0). From the intercepts (-5, 0) and (2, 0), the x-coordinates are -5 and 2, so the zeroes are -5 and 2. Option B is incorrect because the signs are reversed (5 and -2). Options C and D are incorrect because they include 0, which is not the x-coordinate of the given intercepts. Exam tip: read off the x-values of x-axis intersection points to get the zeroes directly.