Geometrical meaning of the zeroes of a polynomial.

Correct answer: B. (b)

Explanation: The zeroes of a polynomial are the x-coordinates of its intersections with the x-axis. The point \((0,0)\) therefore gives the zero 0, while the point \((b,0)\) gives the zero b. The condition \(b\ne0\) ensures that these are two distinct points, so both zeroes must be included in the sum.

Adding the two x-coordinates gives \(0+b=b\). Thus the sum of the zeroes is b, which is option B. The value \(b^2\) would be related to a product in some contexts, not to this sum, and \(-b\) has the wrong sign. The origin should not be ignored: although its x-coordinate is 0, it contributes 0 to the sum while still being an important zero.

Correct answer: B. Below the (x)-axis

Explanation: For a downward-opening parabola, values outside the zeroes are negative. Tip: opening direction changes sign regions.

Correct answer: A. Zero

Explanation: For real (x), (x^6\geq0), so (x^6+1>0). Tip: an always positive polynomial does not cut the (x)-axis.

Correct answer: B. Three

Explanation: A zero of the polynomial is an x‑value where the function equals 0. Here \(p(-7)=0\), \(p(1)=0\) and \(p(4)=0\), so there are three zeroes. The closest incorrect choice "four" is wrong because \(p(-3)=2\), not 0. Exam tip: verify by substituting the x‑value into \(p(x)\) and confirm the result is exactly 0 before counting it as a zero.

Correct answer: A. \((0,0)\) and \((b,0)\)

Explanation: Factor: \(p(x)=x^2-bx= x(x-b)\). x-intercepts occur where \(p(x)=0\), so \(x(x-b)=0\) gives \(x=0\) or \(x=b\). Thus the intercepts are \((0,0)\) and \((b,0)\). The closest distractor (B) is incorrect because \(p(-b)=(-b)^2-b(-b)=2b^2\), which is not zero for general \(b\ne0\). Option (D) is wrong since \((0,b)\) is on the y-axis, not the x-axis. Exam tip: set \(p(x)=0\) and factor out the common \(x\) to find x-intercepts quickly.

Correct answer: A. (10) must be changed to (4)

Explanation: For equal distance from the (y)-axis, zeroes should be opposites, so (4) is needed with (-4). Tip: symmetric zeroes are (a) and (-a).

Correct answer: A. Two points, touching at (x=-2)

Explanation: The zeroes are (-2) and (7), and ((x+2)^2) causes touching at (-2). Tip: the outside (5) does not change the zeroes.

Correct answer: A. Above the (x)-axis

Explanation: In this interval the signs are (+), (-), (-), so the product is positive. Tip: an odd number of negative factors gives a negative value.

Correct answer: A. Because its discriminant is negative

Explanation: The discriminant is (c^2-4c^2=-3c^2<0), so there are no real zeroes. Tip: a negative discriminant means no (x)-axis intersection.

Correct answer: B. One

Explanation: The vertex lies on the (x)-axis, so the parabola touches at ((4,0)). Tip: if the vertex has (y=0), there is one distinct zero.

Correct answer: C. It will not cut

Explanation: The discriminant is (64-88=-24), so there are no real zeroes. Tip: with negative discriminant a parabola does not meet the (x)-axis.

Correct answer: A. (a-3) and (a+3)

Explanation: It is ((x-a)^2-9), so (x-a=\pm3) and the zeroes are (a-3), (a+3). Tip: use difference of squares.

Correct answer: A. 3

Explanation: Mean \(=\dfrac{-1+3+7}{3}=3\). Explanation: The x-coordinates of the intercepts are the zeros; their sum is 9 and dividing by 3 gives the mean 3. Option 9 is incorrect because it is the sum, not the average (division was omitted). Option -3 results from a sign error, and 7 is just one of the zeros. Exam tip: read off the x-values from intercepts first, then compute average = (sum)/(count).

Correct answer: A. (10)

Explanation: 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.

Correct answer: B. Touches at (x=-6) and crosses at (x=2)

Explanation: 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.

Correct answer: B. Below the (x)-axis

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

Correct answer: A. Above

Explanation: 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.

Correct answer: B. \(x=3k\)

Explanation: 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.

Correct answer: A. ((u,0)) and ((v,0))

Explanation: Direct answer: Option A, (u,0) and (v,0). To find x-axis intersections, set p(x)=0. We have x^2-(u+v)x+uv=(x-u)(x-v), because multiplying gives x^2-vx-ux+uv=x^2-(u+v)x+uv. Thus (x-u)(x-v)=0, so x=u or x=v. Each root becomes a point on the x-axis with y=0: (u,0) and (v,0). Option A is correct. Option B gives points on the y-axis, not generally x-axis intersections. Option C changes the signs and is not obtained from the factors. Option D gives general plane points whose y-coordinates need not be zero. If u=v, the two listed points coincide, but the stated locations remain correct. Memory cue: factors (x-a)(x-b) give x-intercepts (a,0) and (b,0).

Correct answer: A. (-5)

Explanation: 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.

Correct answer: A. (-5)

Explanation: 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.

Correct answer: C. Three

Explanation: 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.

Correct answer: B. It will touch at both zeroes

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

Correct answer: A. ((-3,0))

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

Correct answer: A. ((11,0)) and ((-3,0))

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