Geometrical meaning of the zeroes of a polynomial.

We have \(p(x)=x^2-cx\). Factor the expression: \(p(x)=x(x-c)\). Setting \(p(x)=0\) gives \(x=0\) or \(x=c\). Therefore the x-intercepts are \((0,0)\) and \((c,0)\). Why others are wrong: option B uses \(-c\) which is the wrong sign; option C omits the zero at \(x=0\); option D includes \((0,c)\), a point on the y-axis, not an x-intercept. Exam tip: set \(p(x)=0\) and factor out the common \(x\) to find zeros quickly.

For equal distance from the (y)-axis, zeroes should be opposites, so (6) is needed with (-6). Tip: symmetric zeroes are (a) and (-a).

The zeroes are (-3) and (10), and ((x+3)^2) causes touching at (-3). Tip: the outside (7) does not change the zeroes.

In this interval the signs are (+), (-), (-), so the product is positive. Tip: the product of two negative factors is positive.

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

The vertex lies on the (x)-axis, so the parabola touches at ((-5,0)). Tip: if the vertex has (y=0), there is one distinct zero.

The discriminant is (144-204=-60), so there are no real zeroes. Tip: with negative discriminant a parabola does not meet the (x)-axis.

It is ((x-b)^2-16), so (x-b=\pm4) and the zeroes are (b-4), (b+4). Tip: use difference of squares.

Zeroes are the x‑values where the graph meets the x‑axis; here they are -2, 4 and 10. The mean is the sum divided by the count: \(\frac{-2+4+10}{3}=4\). The closest distractor B (12) is the sum of the zeroes, not the mean. Exam tip: read only the x‑coordinates from intercepts and divide by the number of zeroes.

Answer: A, p=5 and q=−36. The x-axis intersections give the roots −9 and 4. Since the coefficient of x² is 1, form the polynomial by multiplying the corresponding factors: (x−(−9))(x−4)=(x+9)(x−4). Expand: x²−4x+9x−36=x²+5x−36. Comparing this with x²+px+q gives p=5 and q=−36. Option A is correct. Option B has the wrong sign for p. Option C uses 13 and 36 without respecting the sum and product signs. Option D incorrectly treats the root sum as p and also gives the wrong sign for q. Check by Vieta’s relations: root sum = −p, so −5=−p and p=5; root product=q, so (−9)(4)=−36. Memory cue: for x²+px+q, sum of roots is −p and product is q.