Geometrical meaning of the zeroes of a polynomial.

The (x)-value of the midpoint is (\frac{-2+10}{2}=4). Tip: on the (x)-axis both points keep (y=0).

The axis of symmetry passes through the average (x=\frac{1+7}{2}=4). Tip: the middle of two zeroes is useful in a parabola.

A non-zero constant multiplier does not change zeroes. Tip: zeroes come from factors that make the value zero.

Factorize: \(x^3-4x = x(x^2-4)=x(x-2)(x+2)\). The x-intercepts (zeros) are values making any factor zero, so \(x=-2,0,2\). Distractor B is wrong because 4 is not a root (\(4^3-4\cdot4=64-16=48\neq0\)). Exam tip: always factor out the common factor \(x\) first, then factor the remaining quadratic as a difference of squares.

A number a is a real zero of a polynomial p(x) if and only if \(p(a)=0\). Therefore 0 is a zero exactly when \(p(0)=0\). On the graph this means the curve must pass through the origin (0,0). The point (5,0) would indicate 5 is a zero, not 0; the point (0,5) has x=0 but y≠0 so 0 is not a root there. Exam tip: compute \(p(0)\) or check the y‑coordinate at x=0 on the graph to decide quickly.

For five distinct real zeroes the degree must be at least (5). Tip: the number of zeroes cannot exceed the degree.

The outside (-2) does not change the zero and ((x+5)^2=0) gives (x=-5). Tip: a squared factor can show touching.

The governing concept is the geometrical meaning of a zero: a real number r is a zero of p(x) exactly when the graph y = p(x) passes through (r, 0). Thus the two points (a, 0) and (b, 0) represent the zeroes a and b, respectively. The condition a < b tells us their order on the number line, so a is the smaller value. Therefore option A is correct. The value b is the larger zero, while 0 is merely the common y-coordinate of both x-axis points and is not automatically a zero. The sum a + b combines the two zeroes but does not identify the smaller one. This reasoning applies regardless of the actual numerical values of a and b.

Zeros of the polynomial are the x-values where p(x)=0. From the table p(-1)=0 and p(3)=0, so the zeros are -1 and 3. Their product is (-1)×3 = −3. Option B (3) ignores the sign; options C and D are incorrect because p(6) and p(-4) are not zero (p(-4)=−2, p(6)=5). Exam tip: only use x-values that satisfy p(x)=0 when identifying roots.

An intersection with the x-axis at \((r,0)\) means the x-coordinate \(r\) is a zero (root) of the graph's equation. Since \(r>0\), the zero is positive. Option A is wrong because \(r\) is not negative; C is wrong because the zero would be 0 only if \(r=0\); D is wrong because an x-axis intersection implies a zero exists. Exam tip: the x-intercept's x-coordinate gives the zero directly.

The governing concept is that x-axis intersections occur where p(x) = 0, and their coordinates are (zero, 0). Factor the polynomial: x² − 3x − 10 = (x − 5)(x + 2), because the two numbers 5 and −2 have product −10 and sum −3. Setting each factor equal to zero gives x − 5 = 0, so x = 5, and x + 2 = 0, so x = −2. Consequently, the graph intersects the x-axis at (5, 0) and (−2, 0), making option A correct. Option B reverses both signs, option C gives y-axis points rather than x-axis points, and option D incorrectly uses the constant and linear coefficients as roots.

The governing concept is the graphical interpretation of zeroes. A real zero is an x-value for which the function value is zero, so it is represented by a point where the graph meets the x-axis. If the graph cuts the x-axis at two different points, the two points have different x-coordinates and therefore give exactly two distinct real zeroes. Hence option A is correct. A graph staying above the x-axis has no real zero, while a graph touching the x-axis at only one point has one distinct real zero, often a repeated root for a quadratic. Intersections with the y-axis do not determine zeroes; they show the value p(0), and an ordinary function can meet the y-axis at only one point.

Both have the same (x)-value (3), so there is one distinct zero. Tip: count a repeated value once for distinct count.

From (x=0) or (x+6=0), the zeroes are (0) and (-6). Tip: if a product is zero then one factor is zero.

The graph does not meet the (x)-axis, so there is no real solution. Tip: (p(x)=0) means an (x)-axis intersection on the graph.

The zeroes are the x-coordinates of the x-intercepts. Here the zeroes are -3, 0 and 6, so their sum is -3 + 0 + 6 = 3. The closest distractor B (6) is wrong because it corresponds to omitting the -3 (i.e., adding only 0 and 6). Exam tip: always include every x-intercept's x-value and keep track of signs when summing roots.

x-intercepts occur where the polynomial equals zero. Factorizing gives \(p(x)=x^2-49=(x-7)(x+7)\). Setting \(p(x)=0\) yields \(x=\pm7\), so the graph meets the x-axis at \((7,0)\) and \((-7,0)\). Option C is a common confusion: \((0,7)\) and \((0,-7)\) are y-intercepts, not x-intercepts. Option B would require zeros at \(x=\pm49\) which correspond to \(x^2-2401\), not our polynomial. Exam tip: to find x-intercepts set \(p(x)=0\) and factor; recognize difference of squares quickly.

When listing distinct zeroes, repeated roots are counted only once. Here 5 is repeated, so the distinct real zeroes are 5 and −1. Option C is incorrect because it lists 5 twice; options B and D are incorrect because each omits one of the values. Exam tip: write the values and remove duplicates (think of a set) to list distinct roots quickly.

The vertex lies on the (x)-axis, so the parabola touches at ((-2,0)). Tip: the opening direction does not change the touching (x)-value.

The zeroes of a polynomial are the x-coordinates where its graph meets or cuts the x-axis. Here, the zeroes are -12 and -4. On the number line, -4 lies to the right of -12, so -4 is greater. Equivalently, among negative numbers, the number with the smaller absolute value is greater. Therefore, option B is correct; 0 and 12 are not the given zeroes.