Geometrical meaning of the zeroes of a polynomial.

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.

The direct answer is A: \(-4\) and \(1\). A zero of a polynomial is an x-value for which the polynomial becomes zero. On a graph, the x-axis has y-coordinate 0, so every point where the graph meets or cuts the x-axis has the form \((x,0)\), and its x-coordinate is a zero. The given points are \((-4,0)\) and \((1,0)\), so the zeroes are \(-4\) and \(1\). Option A is correct because it lists exactly these x-coordinates. Option B changes both signs and gives 4 and -1, which are not the displayed coordinates. Option C includes -4 but incorrectly replaces 1 with 0; 0 would be a zero only if the graph passed through \((0,0)\). Option D includes 1 but incorrectly adds 0. Memory cue: x-axis intersection means read the first coordinate, not the second.

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.

The point \((a,0)\) on the graph means that for x=a the y-value is 0. Since the graph represents y = \(p(x)\), we have \(p(a)=0\). Option B (a) is wrong because the x-coordinate being a does not imply the function value equals a; options C and D are also incorrect. Exam tip: any x-intercept has y = 0, so substitute x = a into \(p(x)\) to get \(p(a)=0\).

A number a is a root of a polynomial exactly when \(p(a)=0\). Here for a=0 we have \(p(0)=4\), which is not zero, so 0 is not a root. Option C describes the correct general condition (the constant term equals \(p(0)\) must be zero), but given \(p(0)=4\) that condition is not met. Option D is incorrect because root-criterion does not depend on the polynomial being linear. Exam tip: To test if x=a is a root, substitute a into \(p(x)\) and check whether the value equals 0.

The x-intercept occurs where \(y=0\). Setting \(3x-12=0\) gives \(3x=12\) and hence \(x=4\), so 4 is correct. Option \(-4\) is wrong because substituting gives \(y=3(-4)-12=-24\), not 0. The nearby distractor 3 also fails since \(y=3(3)-12=-3\). Exam tip: to find x-intercept, set \(y=0\) and solve for \(x\).

The zero is the x-value where the function equals \(y=0\). Set \(y=0\): \(0 = x - 8\), so \(x = 8\). The graph intersects the x-axis at \((8,0)\). The closest distractor \(-8\) is incorrect because it would satisfy \(x+8=0\), not \(x-8=0\). Exam tip: To find zeros, always set \(y=0\) (or the polynomial = 0) and solve for x, then check the intersection point on the x-axis.

The zero (x-intercept) is the x-value where y = 0. Setting \(0 = x + 9\) gives \(x = -9\). Option A (9) is a sign-error; option C (0) is wrong because at \(x=0\) the line gives \(y=9\), not 0; option D (18) is simply incorrect. Exam tip: to find an x-intercept, set \(y=0\) and solve for \(x\).

Points on the x-axis satisfy \(p(x)=0\). So set \(2x+6=0\), giving \(2x=-6\) and \(x=-3\). Therefore the graph cuts the x-axis at \(x=-3\). Option B (3) is a sign error (mistakenly taking \(6/2\) without the negative). Option D (-6) is a distractor and does not make \(p(x)=0\). Exam tip: always find x‑intercepts by solving \(p(x)=0\) first, then check your arithmetic signs.

A point (a, 0) where the graph meets the x-axis means the polynomial p(x) satisfies p(a) = 0. Here p(10) = 0, so 10 is a root (zero) of the polynomial. Option D is incorrect because the constant term is the value of the polynomial at x = 0 (the y-intercept), not the x-coordinate of an x-intercept. Options B and C are wrong since the x-coordinate of the intercept given is 10, not 0 or −10. Exam tip: the x-coordinate of an x-axis intersection gives the root directly — read the coordinate to identify the zero.

A root means the polynomial's value is zero at that x, i.e. the graph meets the x-axis where \(y=0\). The point \((0,7)\) has \(y=7\), so the polynomial is not zero at \(x=0\); therefore \(x=0\) is not a root. The closest wrong distractor is C: lying on the y-axis only indicates \(x=0\), not that \(y=0\). Exam tip: to test a root substitute the x-value into the polynomial and check whether the result is 0 (or check that the graph is on the x-axis).

The zeroes of a polynomial are the x-coordinates of its x-axis intercepts. From the points (-7,0), (0,0) and (4,0) the x-values are -7, 0 and 4, so the zeroes are (-7), (0) and (4). Option B is incorrect because it reverses signs (turning -7 into 7 and 4 into -4) — sign matters for zeros. Option D shows 0 twice, which would indicate multiplicity 2 at x=0, but the given intercepts are distinct so that is not the case. Exam tip: read the x-coordinate of an intercept (x,0) to get the corresponding zero directly; watch the sign carefully.

For (p(x)=-3), the (y)-value is never (0). So the graph does not cut the (x)-axis and has no zero.

A zero of a polynomial is a number that makes the value of the polynomial equal to zero. If substituting a particular number into the variable gives a result of zero, that number has a special name: it is called a zero or root of the polynomial. This idea is also connected with the graph, because the zeroes are the values of the variable where the graph meets the horizontal axis.

Here the condition is already given as \(p(c)=0\). It says that when \(x=c\) is used, the polynomial produces zero. Therefore, \(c\) is the zero of the polynomial. It is not the degree, constant term, or coefficient, because those describe other features of a polynomial. Thus option A is the correct choice.

The point (-8, 0) means y = 0 when x = -8. Since the polynomial's value is y, the polynomial is zero at x = -8 (i.e. f(-8)=0). Option B (x = 8) has the wrong sign; option C (x = 0) would be correct only for the point (0,0); option D is unrelated. Exam tip: an x-intercept (y=0) gives the root directly as the x-coordinate shown.

When a graph crosses the \\(x\\)-axis the output \\(y\\) equals 0. For a polynomial \\(y=p(x)\\), intersections at \\(x=2\\) and \\(x=9\\) mean \\(p(2)=0\\) and \\(p(9)=0\\). Option B and C are incorrect because they assign nonzero values despite the graph meeting the x-axis; option D is incorrect because the intercepts give definite function values (zero). Exam tip: Any x-intercept of a polynomial corresponds to a root—record the function value as 0 immediately.

The zero of a parabola is the x-coordinate where it meets the x-axis. Since the parabola meets (touches) the x-axis only at (-2,0), that x-value is the only real zero; touching means a repeated root (multiplicity 2) at x = -2. Thus the correct zero is -2. Options B and C are incorrect because their x-values do not match the given point; option D is incorrect because touching gives one repeated zero, not two distinct zeros. Exam tip: when a parabola touches the x-axis at its vertex, the x-coordinate of the vertex is the zero (a double root).

If the zeroes are (1) and (-6), the graph cuts the (x)-axis at those (x)-values. So the points are ((1,0)) and ((-6,0)).