Geometrical meaning of the zeroes of a polynomial.

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.

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.

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).

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 \((-4,0)\).

From (x=0) or (x-5=0), we get (x=0,5). Tip: if a product is zero, at least one factor is zero.

A zero of a polynomial is an x-value at which the polynomial value is 0. Geometrically, p(a) = 0 means that the graph contains the point (a, 0) on the x-axis. Here p(1) = 0, p(2) = 0, and p(3) = 0, so the graph contains (1, 0), (2, 0), and (3, 0). Because the question explicitly states that the three zeroes are distinct, these are three different points. Hence option C is correct. Option A and option B count too few intersections. Option D incorrectly counts both coordinates or treats each zero as producing more than one point. Each distinct x-value paired with y = 0 gives exactly one distinct x-axis point.

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

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.

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

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.

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.

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).

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

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.

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.

At both given (x)-values, the function value is (0). Tip: when (p(0)=0), the graph passes through the origin.

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.

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).

The zeroes are (-3), (2), (8), and the smallest is (-3). Tip: a negative number is smaller than a positive number.

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