Geometrical meaning of the zeroes of a polynomial.

The average of the two zeroes is (4), so the other zero is (10). Tip: connect the axis of symmetry with the midpoint of zeroes.

The even-power factor ((x+6)^2) gives touching and the single factor (x-2) gives crossing. Tip: identify graph behavior from the power of the factor.

(x=-1) lies between the zeroes and an upward-opening parabola is below the axis there. Tip: check the sign region between zeroes.

In this interval the first factor is positive and the second is negative, so the outside negative makes the value positive. Tip: check each factor's sign separately.

Factorizing gives \(p(x)=x^2-6kx+9k^2=(x-3k)^2\), so there is a repeated root at \(x=3k\) and the parabola touches the x-axis at that x-value. Alternatively, the discriminant \(D=(-6k)^2-4\cdot1\cdot9k^2=0\) shows a double root. Note: when \(k=0\) all choices collapse to 0, but for a general (nonzero) k the unique touching point is \(x=3k\). Exam tip: spot a perfect square trinomial or check discriminant = 0 to identify a tangent to the x-axis quickly.

It is ((x-u)(x-v)), so the zeroes are (u) and (v). Tip: write each zero as the point ((x,0)).

The average of the two zeroes is (3), so the other zero is (-5). Tip: set (\frac{a+b}{2}) equal to the axis of symmetry.

A zero is an input x for which the polynomial value is exactly zero. Positive or negative values do not represent zeroes; they only show that the graph lies above or below the x-axis at those inputs. Therefore, in a list of function values, we select only the statements written with equality to 0.

Here \(p(-8)=0\), so -8 is a zero, and \(p(3)=0\), so 3 is another zero. The statements \(p(-2)>0\) and \(p(9)<0\) do not add zeroes. Their sum is \((-8)+3=-5\). Therefore option A is correct. A common mistake is to use the signs or to include every listed input instead of checking which function values are exactly zero.

To find where the graph meets the x-axis, solve \(p(x)=0\). For the given polynomial, \(x^3-25x=0\). Taking the common factor x gives \(x(x^2-25)=0\), and the difference of squares factors the second part as \(x(x-5)(x+5)=0\). A product is zero when at least one factor is zero, so the possible x-values are \(0\), 5, and -5.

These three values are distinct, so the graph has three different x-axis intersection points: \((0,0)\), \((5,0)\), and \((-5,0)\). Repeated factors, if present, would affect multiplicity but not create an additional distinct point. Since the question asks for distinct points, the answer is three, which is option C.

Both factors have even powers, so the graph touches at both places. Tip: an even-power zero usually gives touching, not crossing.

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 midpoint is \(\left(\frac{-11+5}{2},0\right)=(-3,0)\). Tip: on the (x)-axis the midpoint has \(y=0\).

The governing concept is the range of a finite set of real numbers, calculated as the greatest value minus the least value. The zeroes given by the x-intercepts are -12, -3 and 8. The greatest zero is 8 and the least zero is -12. Therefore, range = maximum - minimum = 8 - (-12) = 8 + 12 = 20. Hence option C is correct. The value 11 is the difference between -12 and -3, while 15 is the difference between -3 and 12 only by an incorrect sign interpretation. Option D gives -20, but a range is a non-negative spread and cannot be negative. The middle zero does not affect the maximum-minus-minimum calculation.

(x^2-8x-33=(x-11)(x+3)), so the zeroes are (11) and (-3). Tip: form ((x,0)) points from factors.

The point (0,0) means that when \(x=0\), \(y=p(0)=0\). Roots of a polynomial are the x-values for which \(p(x)=0\). Hence \(x=0\) is indeed a root. The closest distractor is C — claiming it is only a y-intercept — which is incorrect here because (0,0) lies on the x-axis as well, so it gives an x-intercept. Exam tip: substitute the x-coordinate of the given point into \(p(x)\); if it yields zero, that x is a root.

(x^2+14x+49=(x+7)^2), so the touching point is ((-7,0)). Tip: in a perfect square, change the sign to get the zero.

The zeroes are (m), (n), (r), so the mean is (\frac{m+n+r}{3}). Tip: take the first coordinate even in symbolic points.

((x-4)^2+4) is always positive, so (p(x)=0) will not occur. Tip: adding a positive number to a square can prevent intersection.

((x+4)^2) is an even-power factor, so the graph touches at (x=-4). Tip: power (2) shows a repeated zero.

The axis of symmetry is at the average of the zeroes, (\frac{(c-7)+(c+3)}{2}=c-2). Tip: the average rule also works with symbols.