Geometrical meaning of the zeroes of a polynomial.

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

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

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

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

(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.

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

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

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.

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.

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

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

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.

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.

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

The zeroes are (-4) and (12), and ((x+4)^2) causes touching at (-4). Tip: the outside (9) 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 (f^2-4f^2=-3f^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 ((12,0)). Tip: if the vertex has (y=0), there is one distinct zero.

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

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

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.

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.

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

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

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