Geometrical meaning of the zeroes of a polynomial.

Correct answer: B. Two

Explanation: Real zeroes are counted from (x)-axis intersections, not from the (y)-axis intercept. Tip: ((0,-20)) does not show a zero.

Correct answer: A. ((-5,0)) and ((5,0))

Explanation: (x^4-625=(x^2-25)(x^2+25)), and the real zeroes are only (\pm5). Tip: (x^2+25) gives no real zero.

Correct answer: C. (7)

Explanation: For seven distinct real zeroes, the degree must be at least (7). Tip: the number of distinct zeroes cannot exceed the degree.

Correct answer: C. The graph does not cut the (x)-axis

Explanation: (x^2+6x+18=(x+3)^2+9), so there is no real zero. Tip: an always positive form gives no intersection.

Correct answer: A. (0,4,6)

Explanation: (x^3-10x^2+24x=x(x-4)(x-6)), so the zeroes are (0), (4), (6). Tip: first take (x) as the common factor.

Correct answer: A. It is the midpoint of the two zeroes

Explanation: (14=\frac{8+20}{2}), so it is the midpoint of the two zeroes. Tip: the midpoint value need not be a zero.

Correct answer: B. Below

Explanation: For (x<-11), both factors are negative and the outside negative makes the value negative. Tip: check factor signs first.

Correct answer: A. -45

Explanation: The x-values where the graph touches or crosses the x-axis are the zeroes. Here the zeroes are 9 and −5, so their product is 9×(−5)=−45. Note: touching usually means an even multiplicity and crossing means an odd multiplicity, but the numerical root values remain 9 and −5; unless the question asks for product counting multiplicities explicitly, use these root values. Option B (45) is the closest distractor but has the wrong sign. Exam tip: remember touch → even multiplicity, cross → odd multiplicity; for product-of-roots questions, multiply the root values given by the graph.

Correct answer: B. \(0\)

Explanation: The x-coordinates of the x-intercepts are the polynomial's zeros: 0, d and −d. Their product is 0\times d\times(−d)=0 because any product containing 0 equals 0. Note that \(d\neq0\) ensures the other two zeros are nonzero, but the presence of the zero root makes the whole product zero. The closest distractor, \(-d^2\), would be the product of d and −d alone and is wrong because it ignores the zero root. Exam tip: always check intercepts for a root equal to 0 first — it instantly gives the product as 0.

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

Explanation: For a downward-opening parabola, values between the two zeroes are positive. Tip: (x=4) lies between the zeroes.

Correct answer: A. Zero

Explanation: For real (x), (x^{10}\geq0), so (x^{10}+1>0). Tip: an always positive polynomial does not cut the (x)-axis.

Correct answer: B. 3

Explanation: A value x is a zero of the polynomial exactly when \(p(x)=0\). Here \(p(-10)=0\), \(p(3)=0\) and \(p(12)=0\), so there are three zeroes. Since \(p(-1)=4\) is not zero, it is not counted. Exam tip: verify the function value equals 0 exactly rather than assuming from sign or proximity.

Correct answer: A. (0,0) and (e,0)

Explanation: Factorize: \(x^2-ex=x(x-e)\). x-axis intercepts occur at x-values where the polynomial equals zero. Thus x=0 and x=e give intercepts \((0,0)\) and \((e,0)\). The closest distractor (option B) is wrong because it uses \(-e\) instead of \(+e\); sign matters when solving \(x-e=0\). Exam tip: always factor out the common factor first and set each factor to zero to find x-intercepts.

Correct answer: A. (18) must be changed to (8)

Explanation: For equal distance from the (y)-axis, zeroes should be opposites, so (8) is needed with (-8). Tip: symmetric zeroes are (a) and (-a).

Correct answer: A. Two points, touching at (x=-4)

Explanation: The zeroes are (-4) and (12), and ((x+4)^2) causes touching at (-4). Tip: the outside (9) 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: the product of two negative factors is positive.

Correct answer: A. Because its discriminant is negative

Explanation: The discriminant is (f^2-4f^2=-3f^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 ((12,0)). Tip: if the vertex has (y=0), there is one distinct zero.

Correct answer: C. It will not cut

Explanation: The discriminant is (256-336=-80), so there are no real zeroes. Tip: with negative discriminant a parabola does not meet the (x)-axis.

Correct answer: A. (c-5) and (c+5)

Explanation: It is ((x-c)^2-25), so (x-c=\pm5) and the zeroes are (c-5), (c+5). Tip: use difference of squares.

Correct answer: A. 5

Explanation: The mean is the average of the x-coordinates of the x-intercepts:
\(\frac{-3+5+13}{3}=\frac{15}{3}=5\). So 5 is correct. Closest distractor D (15) is the sum of the zeros, not the mean; C (13) is just one root; B (−5) reflects a sign error. Exam tip: read the x-values of intercepts first, sum them, then divide by the number of zeros to get the mean.

Correct answer: A. (11)

Explanation: The average of the two zeroes is (5), so the other zero is (11). Tip: the axis of symmetry passes through the midpoint of zeroes.

Correct answer: A. Touches at (x=4) and crosses at (x=-3)

Explanation: An even-power zero gives touching and an odd-power zero gives crossing. Tip: identify graph behavior from the power of the factor.

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

Explanation: (x=-4) lies between the two zeroes and an upward-opening parabola stays below there. Tip: check the sign region between zeroes.

Correct answer: A. Above

Explanation: In this interval the factor signs are (+), (+), (-), and the outside negative makes the value positive. Tip: apply the outside sign at the end.