Exercise 2.1 answers: Polynomials

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(i)

1. The graph is a straight line parallel to the X-axis, drawn above it.
2. It never meets the X-axis, so there is no x for which p(x) = 0.

AnswerNumber of zeroes = 0

(ii)

1. On the left the curve stays below the X-axis: it rises to a small hump and falls to a dip, and neither reaches the axis.
2. After the dip it climbs steeply and cuts the X-axis once, to the right of O.

AnswerNumber of zeroes = 1

(iii)

1. The curve comes up from below and cuts the X-axis once, to the left of O.
2. It rises to a peak, comes back down and cuts the X-axis a second time.
3. It turns at a low point below the axis and climbs through the X-axis a third time.

AnswerNumber of zeroes = 3

(iv)

1. The graph is a U-shaped curve, to the left of O, whose lowest point is just below the X-axis.
2. Both arms of the U cut the X-axis, at two different points.

AnswerNumber of zeroes = 2

(v)

1. Both ends of the curve are below the X-axis, and it moves up and down like a wave.
2. It goes up through the X-axis, down through it, up through it again, and down through it once more: four meeting points.

AnswerNumber of zeroes = 4

(vi)

1. On the far left the curve comes up from below and cuts the X-axis once.
2. It rises to a small hump, comes down and just rests on the X-axis, then rises again without going below it. This touching point is also a zero.
3. It rises to a bigger hump near the Y-axis, comes down, rests on the X-axis once more to the right of O, and then climbs away. That is another zero.
4. So the curve meets the X-axis at 3 points: 1 crossing and 2 touching points.

AnswerNumber of zeroes = 3

Watch this explained “The rests are easy to miss”, 10:18 into Why degree puts a ceiling on how many zeroes there can be