Geometrical meaning of the zeroes of a polynomial.

Correct answer: A. -24

Explanation: A zero of p(x) is an input x for which p(x)=0. From the information given, p(-6)=0 and p(4)=0, so the stated zeroes are -6 and 4. The values p(-1)>0 and p(7)<0 are sign information, not additional zeroes, because neither value equals zero. Their product is (-6)(4)=-24. Therefore Option A is correct. Option B loses the negative sign, Option C incorrectly multiplies or combines the two x-values as a difference, and Option D would be possible only if one of the zeroes were 0. The inequalities help describe signs near the roots but do not change the requested product.

Correct answer: C. Three

Explanation: The governing concept is that distinct real zeroes correspond to distinct points where the graph meets the x-axis. First factor out the common factor x: x^3-16x = x(x^2-16). Then use the difference of squares: x(x^2-16)=x(x-4)(x+4). Setting each factor equal to zero gives x=0, x=4 and x=-4. These are three different real x-values, so the graph has three distinct x-axis intersection points. Option C is correct. Option A omits two roots, Option B omits one root, and Option D incorrectly counts beyond the three factors or treats multiplicity as an additional distinct point.

Correct answer: A. 2 and −6

Explanation: The governing concept is that a zero occurs when at least one factor of a factored polynomial equals zero. From x−2=0, we obtain x=2. From (x+6)^3=0, the base must be zero, so x+6=0 and x=−6. The exponent 3 shows that −6 has multiplicity three, meaning it is repeated three times as a root, but it is still only one distinct zero value. Therefore the distinct zeroes are 2 and −6, making option A correct. Option C lists repetition rather than distinct values, option B changes both signs, and option D omits the zero from the first factor.

Correct answer: C. 16

Explanation: The relevant concept is the range of a set of real numbers, which is the greatest value minus the least value. The zeroes are -10, -4 and 6. The greatest zero is 6 and the least zero is -10. Therefore the range is 6-(-10)=6+10=16. The middle value -4 does not determine the range. Hence Option C is correct. Option A is the magnitude of the smallest zero, and Option B can arise from comparing only -4 and 8 or from an unrelated difference; it is not the full spread. Option D reverses the subtraction and gives a negative value, but a range is never negative.

Correct answer: B. Two

Explanation: The governing concept is the geometrical meaning of a zero: the x-coordinate of a point where the polynomial graph meets the x-axis. The listed x-coordinates are 2, 2 and 9. The repeated point (2,0) represents the same x-value twice, so it contributes only one distinct real zero. The other distinct x-value is 9. Thus the set of distinct zeroes is {2,9}, containing two elements, and Option B is correct. Option A ignores 9, while Option C counts the repeated coordinate twice rather than counting distinct values. Option D is not supported because only two different x-coordinates have been given.

Correct answer: A. p=1, q=-12

Explanation: The governing concept is the relationship between the zeroes and the factorised form of a monic quadratic. Since the graph meets the x-axis at x=-4 and x=3, these are the zeroes. Therefore p(x)=(x+4)(x-3). Expanding gives x^2-3x+4x-12=x^2+x-12. Comparing this with x^2+px+q, the coefficient of x is p=1 and the constant term is q=-12. Hence Option A is correct. Option B has the wrong sign for p, Option C uses the sum with an incorrect sign and the wrong constant sign, and Option D also has the wrong coefficient of x. The leading coefficient is 1, so no additional scaling is needed.

Correct answer: A. It will touch the x-axis at x = 3

Explanation: The governing concept is the geometrical meaning of zeroes: a zero is an x-coordinate where the graph meets the x-axis. Factor the polynomial: p(x) = 3x^2 - 18x + 27 = 3(x^2 - 6x + 9) = 3(x - 3)^2. Thus the only zero is x = 3, and it is repeated. A quadratic with a repeated real zero touches the x-axis at that point instead of crossing it. Therefore, option A is correct. Option B has the wrong sign, option C would require two distinct zeroes, and option D would apply only when there are no real zeroes.

Correct answer: A. 1

Explanation: Answer: A, 1 unit. The zeroes are the x-values where the graph meets the x-axis. Set p(x)=0 and factor the quadratic. We need two numbers with product 42 and sum 13: 6 and 7. Thus x²−13x+42=(x−6)(x−7), so the zeroes are 6 and 7. The corresponding points are (6,0) and (7,0). Their distance on the x-axis is the absolute difference |7−6|=1 unit. Option A is correct. Option B is one zero, not the separation. Option C is the sum of the zeroes, since 6+7=13. Option D is their product, since 6×7=42. Those relationships help with factorisation but do not answer the distance question. Memory cue: distance between two x-axis points is the absolute difference of their x-coordinates.

Correct answer: A. 4 and -7

Explanation: A zero is obtained by setting any factor equal to zero. From x - 4 = 0, we get x = 4. From (x + 7)^3 = 0, we get x + 7 = 0, so x = -7. The exponent 3 tells us that -7 has multiplicity three, meaning it is repeated as a root, but it remains only one distinct zero. Therefore the set of distinct zeroes is {4, -7}, so option A is correct. Option C lists the repeated root several times and therefore describes multiplicity rather than distinct values. Option B changes both signs, while option D incorrectly omits the zero arising from x - 4.

Correct answer: A. It will touch the x-axis at x = 3

Explanation: The relevant principle is that a repeated real zero makes a parabola touch the x-axis, whereas two distinct real zeroes make it cross at two points. Factor the expression: 4x^2 - 24x + 36 = 4(x^2 - 6x + 9) = 4(x - 3)^2. Thus the discriminant is zero and the only zero is x = 3, repeated twice. The graph therefore touches the x-axis at (3, 0). Option A is correct. Option B results from reversing the sign, option C would require a positive discriminant and two different roots, and option D would require a negative discriminant with no real roots.

Correct answer: A. 1

Explanation: Answer: A, 1 unit. A graph meets the x-axis at the zeroes of its polynomial. Solve x²−15x+56=0 by finding two numbers with sum 15 and product 56. These numbers are 7 and 8, so p(x)=(x−7)(x−8). The zeroes are x=7 and x=8, giving points (7,0) and (8,0). Their distance is |8−7|=1 unit. Option A is therefore correct. Option B is only the smaller zero. Option C is the sum of the zeroes, 7+8=15. Option D is their product, 7×8=56. A common mistake is to select the sum or product because they appear in the polynomial; however, the question asks for separation, not a root relationship. Memory cue: first find both roots, then subtract and take the positive magnitude.

Correct answer: A. x = −2

Explanation: Answer: A, x=−2. For a parabola with zeroes r1 and r2, the axis of symmetry passes through the midpoint of the two x-intercepts, so x=(r1+r2)/2. The first zero is −8. The second is 12 more than −8, so r2=−8+12=4. Their midpoint is (−8+4)/2=−4/2=−2. Therefore the axis is the vertical line x=−2. Option A is correct. Option B results from a sign error in the midpoint. Option C is merely the second zero, not the midpoint. Option D is the sum of the zeroes without dividing by 2. Memory cue: the symmetry axis lies halfway between the two roots, not at either root.

Correct answer: A. It will touch the x-axis at x = 5

Explanation: The graph behavior follows from the multiplicity of the real zero. Factor the polynomial: 5x^2 - 50x + 125 = 5(x^2 - 10x + 25) = 5(x - 5)^2. Thus the only zero is x = 5, repeated twice. A parabola with a repeated real zero reaches the x-axis at that point and turns back, so it touches rather than crosses the axis. Therefore option A is correct. Option B uses the wrong sign for the root. Option C would be appropriate if the quadratic had two distinct real roots, and option D would be appropriate only if the quadratic had no real roots. The non-zero factor 5 changes vertical scaling but does not change the zero.

Correct answer: A. 1

Explanation: Answer: A, 1 unit. The graph's zeroes are the x-values for which p(x)=0. Factor the quadratic by finding numbers with product 72 and sum 17: 8 and 9. Thus x²−17x+72=(x−8)(x−9), so the zeroes are 8 and 9. The graph meets the x-axis at (8,0) and (9,0). The distance between them is |9−8|=1 unit. Option A is correct. Option B is one of the zeroes, not the distance. Option C is their sum, 8+9=17. Option D is their product, 8×9=72. These values are useful for identifying the roots but must not be confused with the distance between them. Memory cue: after factorising, distance between real x-axis zeroes is the absolute difference of the roots.

Correct answer: A. x = −1

Explanation: The zeroes of a quadratic polynomial are the x-coordinates where its parabola meets the x-axis. The axis of symmetry of a parabola with two real zeroes lies exactly halfway between those zeroes. The first zero is −10, and the second is 18 greater, so the second zero is −10 + 18 = 8. Their midpoint is (−10 + 8) ÷ 2 = −2 ÷ 2 = −1. Therefore the vertical axis of symmetry is x = −1, making option A correct. The value x = 8 is merely the second zero, not the midpoint. The values x = 1 and x = −8 are neither the average of the zeroes nor the location of the symmetry axis.

Correct answer: A. 3

Explanation: The zeroes of a polynomial are the x-values at which its graph meets the x-axis. Factor the given cubic: p(x) = x³ − 9x² + 18x = x(x² − 9x + 18) = x(x − 3)(x − 6). Hence the three zeroes are 0, 3, and 6. Their mean is the sum divided by the number of zeroes: (0 + 3 + 6) ÷ 3 = 9 ÷ 3 = 3. Therefore option A is correct. The value 6 is one zero, while 9 is the sum of all three zeroes rather than their mean. The original option 18 ÷ 3 was mathematically equal to 6 and could duplicate option B, so it has been corrected to 18, which is the constant term and still remains an incorrect distractor.

Correct answer: A. p = 3, q = −40

Explanation: The x-coordinates of the points where a polynomial graph cuts the x-axis are its zeroes. Thus the zeroes here are −8 and 5. Because the coefficient of x² is 1, the polynomial can be reconstructed as (x − (−8))(x − 5) = (x + 8)(x − 5). Expanding gives x² − 5x + 8x − 40 = x² + 3x − 40. Comparing this expression with p(x) = x² + px + q gives p = 3 and q = −40. Therefore option A is correct. The coefficient p is positive because the middle terms combine to +3x, while q is negative because it equals the product of the zeroes, (−8)(5) = −40.

Correct answer: A. It will touch the x-axis at x = 6.

Explanation: To determine the contact of a quadratic graph with the x-axis, find its zeroes and check whether they are distinct or repeated. Factor the polynomial: 6x² − 72x + 216 = 6(x² − 12x + 36) = 6(x − 6)². Thus the only zero is x = 6, and it occurs twice. A repeated real zero means that the parabola touches the x-axis at one point and turns back, rather than crossing it at two distinct points. Hence option A is correct. The factor 6 only changes the vertical scale of the graph and does not alter the zero. Therefore x = −6 is incorrect, and the graph does meet the axis, so option D is also incorrect.

Correct answer: A. 1

Explanation: The zeroes of the polynomial are the x-coordinates of the graph's intersections with the x-axis. Factor the quadratic by finding two numbers whose product is 90 and whose sum is 19: x² − 19x + 90 = (x − 9)(x − 10). Therefore the zeroes are 9 and 10. Their distance on the x-axis is the absolute difference |10 − 9| = 1 unit, so option A is correct. The value 9 is only one zero, not the separation between the two intercepts. By the coefficient relationships, 19 is the sum of the zeroes and 90 is their product; neither quantity represents the distance. Using the absolute difference also ensures that distance is treated as positive.

Correct answer: A. No zero is shown from the given data

Explanation: The geometrical meaning of a zero is tied specifically to the x-intercept. A number r is a zero of p(x) when p(r) = 0, so the point (r, 0) lies on the graph. The line y = 2 is not the x-axis; every point on it has ordinate 2, not 0. Therefore, its two intersections do not represent zeroes. Since the graph is stated not to cut the x-axis, the supplied information displays no zero. It does not prove that the polynomial has no real zero anywhere unless the whole graph is known, but among the given data, option A is the only certain statement. Options B and C incorrectly treat y = 2 intersections as x-intercepts, while D has no basis.

Correct answer: A. Zero

Explanation: A real zero of a polynomial is a real number r for which p(r) = 0. On the graph y = p(x), this means that the graph must have a point (r, 0), so it must intersect or touch the x-axis. The statement says that the quadratic graph neither touches nor intersects the x-axis. Consequently, there is no real x-coordinate at which p(x) becomes zero, and the number of distinct real zeroes is zero. In discriminant language, a quadratic with no real x-axis contact has discriminant less than zero, although calculating the discriminant is not necessary here. One or two would require contact with the axis; infinitely many is impossible for a nonzero quadratic.

Correct answer: A. Three

Explanation: The x-axis intersections occur where p(x) = 0. Since 5 is non-zero, it cannot itself create a zero; the product is zero when at least one factor is zero. Thus x + 1 = 0 gives x = -1, x - 2 = 0 gives x = 2, and x - 4 = 0 gives x = 4. These are three different real values, so the graph has three distinct x-intercepts: (-1, 0), (2, 0), and (4, 0). Therefore option A is correct. The degree is three, so three distinct zeroes are possible and occur here. Option B or C would require repeated or missing roots, while option D wrongly counts the constant coefficient 5 as an additional zero.

Correct answer: A. 0

Explanation: The governing concept is factorisation and the distinction between real and non-real zeroes. Factor the polynomial as x^4-1=(x^2-1)(x^2+1)=(x-1)(x+1)(x^2+1). The first two factors provide the known real zeroes x=1 and x=-1. For every real number x, x^2 is non-negative, so x^2+1 is at least 1 and can never be zero. Its formal solutions are x=i and x=-i, which are non-real and must not be counted. Therefore there are no real zeroes beyond the two given ones, so option A is correct. Options B, C, and D result from failing to factor completely or from counting complex roots as real roots.