Geometrical meaning of the zeroes of a polynomial.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.