Geometrical meaning of the zeroes of a polynomial.

Correct answer: A. The point where the graph cuts the (x)-axis

Explanation: A zero is the (x)-value for which (p(x)=0). On the graph, it is shown where the curve meets the (x)-axis.

Correct answer: A. 3

Explanation: The x-value where the graph intersects the x-axis is a root of the polynomial because the polynomial's value is zero at that x. Here the graph intersects at x = 3, so the zero is 3. The closest distractor −3 is wrong because the sign is opposite; 0 and 1 do not match the given intersection point. Exam tip: note the difference between 'cuts' and 'touches' — both give zeros, but touching usually indicates even multiplicity while cutting indicates odd multiplicity.

Correct answer: A. (-2) and (4)

Explanation: Zeroes are the (x)-coordinates of the points where the graph meets the (x)-axis. So the zeroes are (-2) and (4).

Correct answer: A. One

Explanation: The graph of a linear polynomial is a line and usually cuts the (x)-axis once. Therefore it has one zero.

Correct answer: A. Where the graph touches or cuts the (x)-axis

Explanation: For a zero, (y=0), and this happens on the (x)-axis. So look at the points where the graph meets the (x)-axis.

Correct answer: A. (0)

Explanation: Real zeroes appear only where the graph meets the (x)-axis. If it does not meet the (x)-axis, it has no real zero.

Correct answer: A. Two

Explanation: The graph of a quadratic polynomial is a parabola. Two distinct intersections with the (x)-axis show two real zeroes.

Correct answer: A. One

Explanation: When a parabola touches the (x)-axis at one point, it has one real zero. In exams, touching the (x)-axis also counts as meeting it.

Correct answer: A. None

Explanation: If the graph does not meet the (x)-axis, no real value satisfies (p(x)=0). Therefore it has (0) real zeroes.

Correct answer: A. Points where the graph meets the x-axis

Explanation: Zeros (roots) are x-values for which \\(p(x)=0\\). On the graph these correspond to points with \\(y=0\\), i.e. where the curve meets the x-axis. So if the curve intersects the x-axis at \\( (a,0) \\), then a is a root of \\(p(x)\\). Option B is incorrect because y-axis intersections give \\(p(0)\\) (x=0) and need not be zero. Option C is incorrect since turning points relate to derivative conditions, not necessarily to the function value being zero. Option D is only true in the special case \\(p(0)=0\\). Exam tip: To read zeros from a graph, list the x-coordinates of all intersections with the x-axis.

Correct answer: A. 0

Explanation: A point on the x-axis has coordinates (x,y) with y = 0. If the graph meets the x-axis at (0,0), the x‑coordinate is 0, so the polynomial satisfies P(0)=0 and 0 is a root. Options B and C would mean x = 1 or x = −1 respectively, which are not indicated by the given point. Option D is incorrect because the x‑value is directly given by (0,0). Exam tip: remember coordinates are (x,y) and any x‑intercept has y = 0, so the x‑value at that point is a root of the polynomial.

Correct answer: A. -5 is a zero of the polynomial

Explanation: The x-coordinate where a graph meets the x-axis is a root of the polynomial because at that point the function value is zero. So if the graph cuts at x = -5, then the polynomial satisfies \(f(-5)=0\) and -5 is a zero. Option B (5) is a common trap — the sign matters, +5 would be a root only if the graph crossed at x = 5. Options C and D contradict the given point of intersection. Exam tip: always read the x-coordinate of the intercept (including its sign); that x-value is the zero of the polynomial.

Correct answer: A. (3)

Explanation: A zero of a polynomial is an x-value at which the graph has y-coordinate 0. On a coordinate plane, every point on the x-axis has the form \((x,0)\). Therefore, each different point where the graph meets the x-axis represents a different zero of the polynomial. The number of zeroes is found by counting these distinct x-axis points, not by counting the y-coordinate, which is 0 for all of them.

The graph meets the x-axis at \((1,0)\), \((2,0)\), and \((5,0)\). Their x-coordinates are 1, 2, and 5, so the zeroes are 1, 2, and 5. Since these are three distinct values, the graph has three zeroes. Thus option A, 3, follows. A repeated point or repeated zero would be counted only once when the question asks for distinct intersections.

Correct answer: A. No zero

Explanation: The graph of (p(x)=5) is a line parallel to the (x)-axis and does not cut it. Hence it has no zero.

Correct answer: A. 4

Explanation: A zero (root) is an x-value where the polynomial equals zero. Set \(p(x)=0\): \(x-4=0\) gives \(x=4\). Thus the graph meets the x-axis at \(x=4\). The nearest distractor \(-4\) is incorrect because \(p(-4)=-4-4=-8\), not zero. Exam tip: to find zeros, always solve \(p(x)=0\) directly rather than inspecting coefficients.

Correct answer: B. -2

Explanation: A zero (root) is the x-value where the polynomial equals zero, i.e. solve \(p(x)=0\). For \(x+2=0\) we get \(x=-2\), so the zero is \(-2\). The common wrong choice 2 is incorrect because \(p(2)=2+2=4\), not zero. Exam tip: substitute the candidate x into the polynomial to verify it yields 0.

Correct answer: A. (3) is a zero of the polynomial

Explanation: The point (3, 0) means that at x = 3 the polynomial's value y = 0, so x = 3 is a root (zero) of the polynomial. Option B is incorrect because 0 would be a zero only if the point were (0, 0). Option C is wrong since a coefficient is a constant multiplying a term, not an x-coordinate of a point on the graph. Option D is wrong because y = 0 does not imply a maximum. Exam tip: zeros of a polynomial are exactly the x-values of its x-intercepts where y = 0.

Correct answer: A. No, because \(y\ne 0\) — to be a zero we must have \(y=0\).

Explanation: A zero (root) is an x‑value where the function's value is \(y=0\) (an x‑intercept). At \((0,4)\) the value is \(y=4\), so \(x=0\) is not a zero. The closest distractor (B) is wrong because \(x=0\) alone does not imply a zero unless \(y=0\). Exam tip: to check if an x‑value is a root, check the corresponding y‑coordinate — it must be zero for a root.

Correct answer: A. Two

Explanation: Each distinct intersection with the (x)-axis gives one real zero. With two intersections, there are two real zeroes.

Correct answer: A. When (p(2)=0)

Explanation: A value is called a zero only when the polynomial value at that point is (0). So for (x=2), (p(2)=0) is required.

Correct answer: A. \((-1,0)\)

Explanation: Here \(p(x)\) denotes the polynomial's value (the y‑coordinate). \(p(-1)=0\) means at \(x=-1\) the value is \(y=0\), so the graph passes through the point \((-1,0)\). Option B \((1,0)\) would be correct only if \(p(1)=0\) — it's a sign‑error distractor. Options C and D correspond to values at \(x=0\) and are irrelevant unless \(p(0)=-1\) or \(p(0)=1\) were given. Exam tip: whenever \(p(a)=0\), the x‑coordinate of the zero is \(a\) and the point is \((a,0)\).

Correct answer: A. (6,0)

Explanation: \(p(6)=0\) means the function value is zero at \(x=6\), so the graph meets the x-axis (where \(y=0\)) at the point \((6,0)\). The closest distractor \((-6,0)\) is a sign error — nothing in the given information implies \(p(-6)=0\). Exam tip: whenever you are given \(p(a)=0\), the corresponding x-intercept is always \((a,0)\).

Correct answer: A. (2) and (7)

Explanation: The direct answer is option A: 2 and 7. A zero of a polynomial is an x-value for which the polynomial value is zero. On a graph, this means the point lies on the x-axis, whose y-coordinate is 0. The graph meets the x-axis at (2, 0) and (7, 0), so the corresponding x-coordinates are 2 and 7. Therefore the two zeroes are 2 and 7. Option A is correct because it lists exactly those x-coordinates. Option B is wrong because changing both signs is not justified; the graph gives positive x-coordinates. Option C is wrong because 0 is not one of the stated x-coordinates, even though the points have y-coordinate 0. Option D is wrong because it includes 0 instead of the second x-coordinate 7. A common beginner mistake is to read the zero as the second coordinate; actually, the zero is the x-coordinate of an x-axis intersection. Memory cue: at a point (r, 0) on the x-axis, r is the zero.

Correct answer: A. (4)

Explanation: When a parabola touches the (x)-axis at its vertex, it has one real zero. Its (x)-coordinate is (4).

Correct answer: A. One that cuts the (x)-axis at ((-3,0)) and ((2,0))

Explanation: If the zeroes are (-3) and (2), the graph cuts the (x)-axis at those (x)-values. So the points are ((-3,0)) and ((2,0)).