The three symmetric relations that hold for a cubic

What to assume they know

• What the sum and product of two zeroes reveal about a, b and c — the two relations for a quadratic, and the expansion that forces them
• Why degree puts a ceiling on how many zeroes there can be — that a cubic carries at most three zeroes
• Substituting a number into a polynomial, and what makes it a zero — substituting a number and testing whether the output is 0
• The general cubic ax³ + bx² + cx + d with a ≠ 0, from Degree as the label that separates linear, quadratic and cubic
• Arithmetic with fractions, including cubing a negative fraction

What they should be able to do

• Verify that a stated number is a zero of a cubic by substitution, showing the working
• Compute the three combinations of three zeroes: their sum, the sum of their products in pairs, and their product
• Match each combination against the corresponding ratio of coefficients, including the sign
• Derive all three relations by expanding the product of three linear factors and matching coefficients against ax³ + bx² + cx + d
• Explain why the signs alternate, in terms of how many factors each term takes its minus sign from
• Explain why a cubic has three relations where a quadratic had two
• Say what must be established before the relations may be checked, and why the order matters
• State what the chapter's own footnote means for how this material is assessed

Where it usually goes wrong

• "There are three relations because there are three coefficients to match." There are four coefficients in a cubic. There are three relations because there are three ways of combining three zeroes symmetrically, and the leading coefficient is used up fixing the scale.
• "The minus signs are arbitrary and must be memorised." They alternate, and the expansion says why: a term built from an odd number of zeroes carries an odd number of minus signs. Once a student sees that, the pattern extends to any degree without further memorising.
• "Two at a time means multiply the first two." It means all three distinct pairs, and the products of all three pairs are added. Draw the three pairs explicitly.
• "The product of the zeroes is d/a." For a cubic it is −d/a, where for a quadratic the product was c/a with no minus. Set the two side by side and let the alternation explain the difference, rather than treating them as two unrelated formulas.
• "You can check the relations on any three numbers." Not until they have been shown to be zeroes. Example 5 spends its first half on exactly that, and the habit is what the assessment is really testing.
• "The chapter shows how to find a cubic's zeroes." It does not. Both worked cubics arrive with their zeroes supplied, and the task is verification. An explanation that implies otherwise sets up a method that does not exist in this book.
• "Example 5 is off the syllabus, so the cubic relations are too." The footnote is attached to the worked example, and the three relations survive into the chapter's Summary.

Questions to check understanding

• Verify that stated numbers are the zeroes of a stated cubic, then check all three relations — the exact shape of Example 5
• Given the three zeroes, write down a cubic that has them
• Given two zeroes of a cubic and its coefficients, find the third using the sum or the product relation
• Find an unknown coefficient from a stated condition on the zeroes
• State all three relations for a general cubic from memory, with correct signs
• Explain why the product relation carries a minus sign for a cubic but not for a quadratic

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated polynomials.

• The specimen cubic (§2.3, p. 21): p(x) = 2x³ − 5x² − 14x + 8, offered with the three numbers 4, −2 and 1/2. The section invites the reader to check them and observes that since a cubic cannot carry more than three, these must be all of them. Verified by substitution: p(4) = 128 − 80 − 56 + 8 = 0; p(−2) = −16 − 20 + 28 + 8 = 0; p(1/2) = 1/4 − 5/4 − 7 + 8 = 0.
• The three combinations for that cubic (§2.3, p. 21). Here a = 2, b = −5, c = −14, d = 8. Verified:
  • sum: 4 + (−2) + 1/2 = 5/2, and −b/a is 5/2;
  • two at a time: 4(−2) + (−2)(1/2) + (1/2)(4) = −8 − 1 + 2 = −7, and c/a is −14/2 = −7;
  • all three: 4 × (−2) × 1/2 = −4, and −d/a is −8/2 = −4. The middle one is the new arrival and deserves the most attention: there are three ways to pick two zeroes out of three, and all three products are added.
• The expansion that proves it, which the chapter does not print. Take α, β and γ as the zeroes. Verified by multiplying out: (x − α)(x − β)(x − γ) = x³ − (α + β + γ)x² + (αβ + βγ + γα)x − αβγ. Multiplying through by a and matching against ax³ + bx² + cx + d gives b = −a(α + β + γ), c = a(αβ + βγ + γα) and d = −aαβγ, which rearrange into the three relations. Note: each term of the expansion picks x from some brackets and the zero from the others, so a term formed from j zeroes carries j minus signs — which is exactly why the signs alternate.
• The three relations as the chapter states them (§2.3, pp. 21–22). For ax³ + bx² + cx + d: the sum of the zeroes equals −b/a; the sum of their products taken two at a time equals c/a; the product of all three equals −d/a.
• Example 5 (§2.3, p. 22): p(x) = 3x³ − 5x² − 11x − 3, with 3, −1 and −1/3 offered as the zeroes. Coefficients a = 3, b = −5, c = −11, d = −3. Verified, substitution first: p(3) = 81 − 45 − 33 − 3 = 0; p(−1) = −3 − 5 + 11 − 3 = 0; p(−1/3) = −1/9 − 5/9 + 11/3 − 3, which is (−1 − 5 + 33 − 27)/9 = 0. Verified, relations second: sum 3 − 1 − 1/3 = 5/3, and −b/a is 5/3; two at a time, 3(−1) + (−1)(−1/3) + (−1/3)(3) = −3 + 1/3 − 1 = −11/3, and c/a is −11/3; product 3 × (−1) × (−1/3) = 1, and −d/a is 3/3 = 1.
• The order the example insists on. Example 5 does the substitutions before it touches the relations. That is not padding: the relations are statements about the zeroes, so a set of numbers that had not been shown to be zeroes would make the second half meaningless. Build the explanation the same way round.
• The printed footnote on Example 5 (§2.3, p. 22). The example carries a footnote putting it outside the examination. The relations themselves are not exempted — §2.4 keeps all three as its final summary point.
• The third Greek letter. The footnote on p. 19, in the quadratic derivation, announces in advance that a third letter will be needed later. It arrives on p. 21. Small, but it is the chapter telling the reader that the quadratic case was a special case of something wider.

Figures to have open

• A three-dot diagram of the zeroes with the three connecting pairs drawn in, so that "two at a time" is something seen rather than parsed. Not in the book; this is an added figure and it carries section 5.
• The expansion of the product of three linear factors laid out so that terms can be grouped by how many zeroes they contain. Not in the book — §2.3 states that the relations can be proved and does not print the working.
• A side-by-side card of the quadratic relations and the cubic relations with the signs aligned. Standard schematic.
• No textbook art is required — §2.3 carries no figures.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 2 "Polynomials", §2.3, whose printed heading is "Relationship between Zeroes and Coefficients of a Polynomial", pp. 21–22 — from the specimen cubic through the three general relations and Example 5 with its footnote.
• The chapter's §2.4 "Summary", p. 23, keeps all three relations as its sixth and final point.
• Backward pointer inside the same chapter: the footnote announcing the third Greek letter is at §2.3, p. 19.