Geometrical meaning of the zeroes of a polynomial.

Correct answer: A. The graph may cross the x-axis between x = 2 and x = 5

Explanation: Polynomial functions are continuous: their graphs have no jumps or breaks. Since p(2) is positive, the graph is above the x-axis at x = 2; since p(5) is negative, it is below the x-axis at x = 5. Moving continuously from x = 2 to x = 5, the graph must pass through y = 0 at least once. Thus it has at least one real zero in the interval (2, 5), so it may cross the x-axis between those values. The wording “may cross” is appropriate because the information does not determine the exact zero or exclude multiple crossings. Options B and C are unsupported, and a polynomial graph cannot be a vertical line parallel to the y-axis.

Correct answer: C. 2

Explanation: By definition a number \(a\) is a zero of the polynomial only if \(p(a)=0\). Here \(p(2)=0\) is given, so 2 is definitely a root. The statements \(p(-1)<0\) and \(p(0)<0\) show values that are nonzero (negative), so \(-1\) and \(0\) are not roots. 'None' is incorrect because \(p(2)=0\) explicitly gives a zero. Exam tip: to confirm a zero, look for an explicit equality \(p(a)=0\); sign information alone (positive/negative) does not prove a root.

Correct answer: A. Both are 0

Explanation: The graph of y = p(x) intersects the x-axis at a point whose y-coordinate is zero. At (-2, 0), the y-coordinate gives p(-2) = 0. Similarly, at (7, 0), the y-coordinate gives p(7) = 0. Therefore both values are zero, and -2 and 7 are the two zeroes of the quadratic polynomial. The signs of the polynomial between or outside the zeroes do not change these endpoint values.

Correct answer: C. 12

Explanation: Distance on the number line is the absolute difference between the coordinates, so it is always nonnegative. Here distance = \\(|9-(-3)| = |12| = 12\\). Option B (9) is just one zero's x-value, not the distance; option D (-12) is negative and therefore not a valid distance. Exam tip: always use |x2 - x1| to find distance between two zeros on the x-axis.

Correct answer: A. ((2,0))

Explanation: The midpoint of ((-2,0)) and ((6,0)) is ((2,0)). Tip: when (y)-values are equal, average the (x)-values.

Correct answer: C. (r + s) / 2

Explanation: The governing concept is that the zeroes of a polynomial are the x-coordinates of the points where its graph meets the x-axis. From the intersections (r, 0) and (s, 0), the two zeroes are r and s. The average of two values is their sum divided by 2, so average = (r + s) / 2. Therefore, option C is correct. Option A is the product of the zeroes, not their average. Option B gives only their sum and omits division by the number of values. Option D uses a difference, which is unrelated to the usual average formula. The y-coordinate 0 is important geometrically because it identifies the points as x-axis intersections, but it does not enter the average calculation.

Correct answer: B. They are opposites of each other

Explanation: The x-intercepts give the zeros of the polynomial; here the zeros are -4 and 4. They have equal magnitude and opposite signs, so they are opposites of each other. In fact their sum is zero: \(\alpha+\beta=0\) (with \(\alpha=-4,\beta=4\)). The closest distractor A is wrong because "both equal" would mean the same number (e.g. 4 and 4), which is not the case. Exam tip: read the x-coordinates of intercepts as the zeros and compare their signs and magnitudes directly.

Correct answer: B. 0, 2 and 5 are zeros

Explanation: If a graph meets the x-axis at (a,0), then x=a is a root of the polynomial because the polynomial's value is 0 there. Since the graph passes through (0,0), (2,0) and (5,0), the polynomial equals zero at x=0, x=2 and x=5; hence all three are zeros. Closest distractor: option D wrongly omits x=0; options A and C ignore the given x-intercepts. Exam tip: any x-intercept (a,0) directly gives a root x=a — never overlook the origin (0,0).

Correct answer: B. -4

Explanation: The zero is the x-value where y = 0. Set \(y=0\) in \(y=-3x-12\): \(-3x-12=0\) ⇒ \(-3x=12\) ⇒ \(x=-4\). So the zero is \(-4\). The distractor \(4\) is wrong due to the sign; \(12\) and \(-12\) do not satisfy the equation. Exam tip: to find the x-intercept of a line set \(y=0\); here the x-intercept is \((-4,0)\).

Correct answer: A. (0) and (5)

Explanation: The direct answer is option A: the graph meets the x-axis at x=0 and x=5. The x-axis has y=0, so we set p(x)=0. Here 0=x(x-5). A product is zero when at least one factor is zero. Therefore x=0 or x-5=0. Solving the second equation gives x=5. The intersections are consequently (0,0) and (5,0). Option A is correct. Option B includes 1, but x=1 gives p(1)=1(-4)=-4, not 0. Option C uses -5 incorrectly; x=-5 gives a nonzero product. Option D is wrong because both factors produce zero at different x-values, so there are two intersections, not only one. Memory cue: for a factored polynomial, set each factor equal to zero.

Correct answer: C. They are equally distant on both sides of the (y)-axis

Explanation: (-2) and (2) are equally distant from the (y)-axis. Tip: opposite zeroes may indicate symmetry in the graph.

Correct answer: B. Because p(2) ≠ 0

Explanation: The defining condition for x = a to be a zero of p(x) is p(a) = 0. The graph crosses the x-axis at x = 1 and x = 4, so those values correspond to zero function values. At x = 2, however, the question states p(2) > 0. A positive number is not zero, so the graph point at x = 2 has a positive y-coordinate and lies above the x-axis. Therefore x = 2 is not a zero, making option B correct. Being between two zeroes does not automatically make a number another zero. The fact that 2 itself is positive is irrelevant; the decisive issue is the value of p(2). Option D is false because the graph can certainly contain a point with x = 2 and positive y-coordinate.

Correct answer: A. One

Explanation: There is only one intersection, so there is one real zero. Tip: a cubic can have at most three real zeroes, not necessarily three.

Correct answer: C. It has no real zero

Explanation: The graph does not meet the (x)-axis, so (p(x)=0) does not occur. Tip: staying above or below without meeting gives no intersection.

Correct answer: C. (7,0)

Explanation: If x = 7 is a zero of the polynomial p(x), then p(7) = 0. Points on the graph are of the form (x, p(x)), so at x = 7 the y-value is 0 and the point (7,0) must lie on the graph. The distractor (7,7) is wrong because it gives y ≠ 0; (0,7) is wrong due to reversed coordinates; (-7,0) is wrong because the x-coordinate is -7, not 7. Exam tip: put the zero as the x-coordinate and set y = 0 to get the point.

Correct answer: B. The zeroes are \(a,b,c\)

Explanation: An x-axis intercept (x,0) means the function value is zero at that x, so the x-coordinates a, b, c of (a,0), (b,0), (c,0) are the zeroes. Option A is wrong because (0,0,0) does not represent the three x-values a, b, c. Option C is false since (a+0)=a (and similarly for b, c), so writing (a+0,b+0,c+0) is equivalent to (a,b,c). Option D is wrong because all three a, b and c are zeroes, not just c. Exam tip: whenever you see an intercept (p,0), the zero of the polynomial/function is p (the x-coordinate).

Correct answer: A. It will touch at (x=3)

Explanation: ((x-3)^2=0) gives only (x=3), and the square causes touching. Tip: the outside (2) does not change the zero.

Correct answer: A. 5 and -5

Explanation: Set p(x)=0 to find zeros:
\(-x^2+25=0\) gives \(x^2=25\), so \(x=\pm5\). Thus the real zeros are 5 and −5. Option B is incorrect because it confuses the value of \(x^2\) (25) with the values of x; the roots are the square roots, \(\pm5\). Option C is wrong since p(0)=25, not 0. Option D is wrong because real zeros do exist. Exam tip: always solve p(x)=0 (or set y=0) and then solve the resulting equation step by step.

Correct answer: C. 8 units

Explanation: Both points lie on the x-axis, so their distance is found by taking the absolute difference of their x-coordinates. Thus distance = |6 - (-2)| = |6 + 2| = 8 units. A distance cannot be negative, so the sign of the subtraction must be handled using the absolute value. Therefore, the correct answer is 8 units; the other numerical choices do not represent the separation of the two points.

Correct answer: A. The graph passes through ((0,0)) and ((5,0))

Explanation: The direct answer is option A: the graph passes through (0, 0) and (5, 0). The statement p(0) = 0 means that when x = 0, the y-value p(x) is also 0. Therefore the graph contains the point (0, 0), the origin. Similarly, p(5) = 0 means that when x = 5, the y-value is 0, so the graph contains (5, 0). Both points lie on the x-axis, so they represent real zeroes 0 and 5. Option A is correct. Option B is wrong because (0, 5) would mean p(0) = 5, and (5, 5) would mean p(5) = 5, opposite to the given values. Option C is wrong because the graph does meet the x-axis at the stated points. Option D is wrong because it meets the x-axis at x = 5 as well as meeting the y-axis at the origin; it is not limited to the y-axis. Strictly, the information proves that the graph passes through these points; whether it crosses or merely touches at either point is not needed. Memory cue: p(a) = 0 always gives the graph point (a, 0).

Correct answer: B. Two

Explanation: Crossing the x-axis at a point indicates a root with odd multiplicity (a simple real zero); crossing at (2,0) gives the root x=2. Touching the axis indicates a root with even multiplicity; touching at (−1,0) gives the root x=−1. These are two distinct x-values, so there are 2 distinct real zeros. Why "three" is wrong: touching does not create an extra distinct x-value, it only implies even multiplicity at x=−1. Exam tip: count distinct x-values of intercepts; use multiplicity only to determine crossing vs touching behavior.

Correct answer: C. There is no real zero

Explanation: The graph does not meet the (x)-axis and the given values are not zero. Tip: do not call a value a zero unless the function value is (0).

Correct answer: A. (-3)

Explanation: The direct answer is A: -3. The x-axis intersections give the zeroes directly: -3, 2, and 8. To find the smallest number, compare them on the number line. Every negative number is less than every positive number, so -3 is less than 2 and 8. Option A is correct. Option B, 2, is a zero but not the smallest because it is greater than -3. Option C, 8, is also a zero but is the greatest of the three, not the smallest. Option D, 0, is not an x-coordinate of any given intersection, so it is not a zero identified by the data. A common mistake is to choose the number with the smallest-looking positive size or to ignore the negative sign. On a number line, values farther left are smaller. Thus the order is -3, 2, 8.

Correct answer: C. Both (4) and (-6) are zeroes

Explanation: In both touching and crossing, the graph meets the (x)-axis. Tip: for a zero, meeting the axis is enough, crossing is not necessary.

Correct answer: A. On the x-axis the polynomial's value is zero, i.e. \(p(x)=0\)

Explanation: An intersection with the x-axis means y=0, so the x-coordinate satisfies \(p(x)=0\); hence x-axis intercepts are the polynomial's zeros (roots). An intersection with the y-axis occurs at x=0 and the y-value is \(p(0)\), which is not necessarily zero. Thus B is incorrect because it wrongly claims the y-axis always gives \(p(x)=0\). C is only true in the special case that \(p(0)=0\) (i.e. zero is a root); D is false because an x-axis intersection does indicate a zero. Exam tip: set y=0 and solve \(p(x)=0\) for roots; set x=0 to find the y-intercept \(p(0)\).