Why degree puts a ceiling on how many zeroes there can be

What to assume they know

• Which way a parabola opens, and the three ways it can meet that axis — the three cases for a quadratic, and the ceiling of two they establish
• A zero is exactly a point where the curve meets the horizontal axis — that a zero and a crossing are the same thing
• Degree as the label that separates linear, quadratic and cubic — degree, and that multiplying expressions adds their degrees
• Taking a common factor out of an expression, e.g. recognising that x³ − x² has x² in every term
• Substituting negative values into a cube, and getting a negative result

What they should be able to do

• Build a table of values for a cubic and locate the inputs that return 0
• Count the crossings on a supplied cubic curve and match them against the table
• Produce cubics with three, two and one distinct zeroes, and say what makes the difference
• Explain, using factors, why the count of zeroes cannot exceed the degree
• Distinguish a ceiling from a count, and give a printed example that falls short of its ceiling
• State the general remark for degree n and apply it to predict a maximum
• Read the number of zeroes off an unlabelled graph, including cases where the curve touches the axis rather than crossing it

Where it usually goes wrong

• "A cubic has three zeroes." The chapter draws three cubics deliberately and they carry three, one and two. Degree fixes the ceiling, never the count.
• "At most n really means exactly n once the polynomial is nice enough." y = x³ is about as simple as a cubic gets and it has one zero.
• "A curve that flattens as it meets the axis must have extra zeroes there." Fig. 2.7 flattens at the origin and still passes through at a single point. The flattening is a fact about the factor being used more than once, not about extra crossings.
• "A touch does not count, because the curve did not get to the other side." It counts as a meeting point, and it is exactly the situation Fig. 2.4 named for quadratics. In Fig. 2.10 panel (vi) two of the three contacts are touches, and missing them is the standard error on this exercise.
• "You can tell the degree from the picture." You cannot. Fig. 2.10 panel (i) is consistent with a degree-0 polynomial, and a curve with three crossings could belong to a polynomial of degree 3, 4, 5 or higher. The Remark runs from degree to a ceiling on crossings, never backwards.
• "Zeroes have to be found before they can be counted." Every graph question in this section is answered by looking, without solving anything.

Questions to check understanding

• Count the zeroes on a supplied graph — the exact form of Exercise 2.1
• Explain why a given count is what it is, which is what Example 1 asks four times with a one-word prompt
• State the maximum number of zeroes for a polynomial of stated degree
• Given a factored polynomial, list its distinct zeroes and say how many the degree would have allowed
• Produce a cubic with a stated number of distinct zeroes
• Decide whether a stated curve could be the graph of a polynomial of stated degree, and justify

Examples worth working on the board

Values marked verified are worked out here or an added reading of the printed pages.

• Table 2.2 (§2.2, p. 15), for y = x³ − 4x. Inputs −2, −1, 0, 1, 2 against outputs 0, 3, 0, −3, 0. Verified every entry: at −2, −8 + 8 = 0; at −1, −1 + 4 = 3; at 0, 0; at 1, 1 − 4 = −3; at 2, 8 − 8 = 0. Three of the five inputs return 0, which is the point.
• Fig. 2.6 (§2.2, p. 16). The curve of y = x³ − 4x, with four points labelled inside the artwork, read off the printed page: (−2, 0), (−1, 3), (2, 0) and (1, −3). The curve rises, turns, falls, turns again and rises. Verified: the three points sitting on the axis correspond to inputs −2, 0 and 2 — the same three the table found. The chapter states explicitly that the curve meets the axis at these three and nowhere else.
• Fig. 2.7 (§2.2, p. 16), the curve of y = x³. Labelled points read off the page: (2, 8), (1, 1), (−1, −1), (−2, −8). Verified: the only input returning 0 is 0, so a cubic can carry just one zero. The curve flattens as it passes through the origin but does pass through.
• Fig. 2.8 (§2.2, p. 16), the curve of y = x³ − x². Labelled points read off the page: (2, 4), (1, 0), (−1, −2). Verified: 8 − 4 = 4 and −1 − 1 = −2, so the labels are consistent. The chapter factors the polynomial as x²(x − 1), which hands you the two zeroes directly. Verified: the value is 0 exactly when x² = 0 or x − 1 = 0, that is at 0 and at 1 — two distinct zeroes for a cubic. Note: the factor x is used twice here, which is why a degree-3 polynomial has run out of room after two distinct values.
• The factor argument, which the chapter does not spell out. If k is a zero then (x − k) divides the polynomial. Distinct zeroes give distinct factors, and the product of those r distinct factors therefore divides the polynomial, and a divisor cannot outrank what it divides — that is the step the argument turns on and the one most easily skipped. Since the product already has degree r, the polynomial's degree is at least r, so r cannot exceed the degree n. Verified against the chapter's own three cubics: x³ − 4x factors as x(x − 2)(x + 2), three factors and three zeroes; x³ is x·x·x, one distinct factor and one zero; x³ − x² is x·x·(x − 1), two distinct factors and two zeroes. All three sit at or under the ceiling of 3, and only the first reaches it.
• The Remark (§2.2, p. 17). Stated for a polynomial of degree n: the curve can meet the horizontal axis at n points at the outside, so the count of zeroes is capped at n. Note honestly that the chapter asserts this and does not prove it; the factor argument above is what backs it.
• Example 1 (§2.2, pp. 17–18, with Fig. 2.9). Six unlabelled curves, panels (i) to (vi), each said to belong to some polynomial. Read off the printed page: (i) a curve rising from below left, crossing the axis once, then levelling and rising again — one crossing. (ii) an arch whose two ends dip below the axis — two crossings. (iii) a curve crossing, rising to a peak, falling below the axis, then rising through it again — three crossings. (iv) a straight line cutting the axis once. (v) an upward-opening curve whose lowest part rests on the axis — one meeting point, a touch rather than a cut. (vi) a wave crossing the axis four times. The chapter prints the counts 1, 2, 3, 1, 1 and 4 and asks the student to supply the reason for the last four.
• Exercise 2.1 (p. 18, with Fig. 2.10). Six more unlabelled curves. Read off the printed page, with panels (iii), (v) and (vi) enlarged because the distinction between touching and crossing is not resolvable at page scale: (i) a straight line drawn parallel to the horizontal axis and clear of it — it never meets the axis at all. (ii) a curve running below the axis on the left, dipping, then climbing steeply and cutting the axis once on the right. (iii) a curve rising through the axis, peaking, falling through the axis to a low turn, then rising through it again — three crossings. (iv) an upward-opening curve whose lowest part is below the axis, so both arms cut it — two crossings. (v) a wave that cuts the axis, peaks, cuts again, dips below near the origin, cuts a third time, peaks, and cuts a fourth — four crossings. (vi) a curve that cuts the axis on the left, rises to a small peak, comes back down to rest on the axis, rises to a larger peak, comes down to rest on the axis a second time, then rises — one cut and two touches, so three points of contact. Verified as counts, by me, from the enlargements: 0, 1, 3, 2, 4 and 3. The chapter prints no answers.

Figures to have open

• Fig. 2.6, Fig. 2.7 and Fig. 2.8 redrawn as schematics with their printed point labels. These are the chapter's own figures (p. 16); redraw rather than reproduce the printed art.
• Fig. 2.9 redrawn — all six panels, unlabelled as printed, since the exercise is to count from the shape alone (pp. 17–18).
• Fig. 2.10 redrawn — all six panels, with panels (iii), (v) and (vi) available at larger size, because at printed scale a touch and a near-miss look identical (p. 18).
• A factor-to-zero schematic: a product of linear factors with each factor's zero marked on an axis beneath it. Not in the book; this is an added figure and it carries section 7.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 2 "Polynomials", §2.2, pp. 15–18 — the cubic discussion with Table 2.2 and Figs. 2.6, 2.7 and 2.8, the Remark on p. 17, Example 1 with Fig. 2.9, and Exercise 2.1 with Fig. 2.10.
• Backward pointer inside the same chapter: the ceiling of two for a quadratic was reached at §2.2, p. 15.
• The chapter's §2.4 "Summary", p. 23, keeps the ceilings of 2 and 3 as its fourth point but does not restate the general Remark for degree n.