What the sum and product of two zeroes reveal about a, b and c

What to assume they know

• Substituting a number into a polynomial, and what makes it a zero — a zero as an input at which the polynomial returns 0
• Degree as the label that separates linear, quadratic and cubic — the general quadratic ax² + bx + c and why a ≠ 0
• Factorising a quadratic by splitting the middle term, from Class IX — the section assumes it and does not re-teach it
• That a product is 0 exactly when one of the things multiplied is 0
• The identity for a difference of two squares, used in one worked example
• Arithmetic with fractions and with square roots

What they should be able to do

• Factorise a quadratic by splitting the middle term, and read its zeroes off the factors
• Compute the sum and product of two zeroes and compare each against a ratio of coefficients
• Derive both relations by expanding a constant times the product of the two linear factors and matching coefficients
• Explain where the extra constant comes from and why it cannot simply be discarded
• Verify the two relations on a quadratic whose zeroes are irrational
• Construct a quadratic when the two zeroes are unknown but their sum and their product have been prescribed
• Explain why that construction has infinitely many answers, and describe them all
• State what the derivation assumes, and hence when the relations have something to be about

Where it usually goes wrong

• "The sum of the zeroes is b/a." The minus sign is the whole difficulty. Track it back through b = −a(α + β) rather than memorising, and check it on x² + 7x + 10, where both zeroes are negative and b is positive.
• "The relations only work when the zeroes are whole numbers." Example 3 runs on √3 and −√3, and §2.3's second opening specimen, 3x² + 5x − 2 on p. 19, runs on the fraction 1/3. (Not Example 4 — that one is built from sum −3 and product 2, and the polynomial it produces has the whole-number zeroes −1 and −2.) Nothing in the derivation ever assumed anything about what kind of numbers α and β are.
• "You have to find the zeroes before you can use the relations." The whole point is the reverse: the relations give you the sum and the product straight off the coefficients, without solving anything.
• "There is one quadratic with a given sum and product." There is a family, every member a real multiple of the simplest one. The section says so explicitly at Example 4.
• "k is just a tidy-up constant." It is the leading coefficient. Naming it as such is what turns the derivation from bookkeeping into an explanation.
• "Splitting the middle term is a rule about the number in the middle." The pair being hunted must multiply to the product of the outer two coefficients and add to the middle one. Both conditions, every time.
• "If a quadratic has no zeroes, the relations are false." They were derived on the assumption that α and β exist. Where they do not, there is nothing for the relations to be about, and the chapter does not claim otherwise.

Questions to check understanding

• Find the zeroes of a stated quadratic and check both relations against its coefficients — the exact form of Exercise 2.2 item 1
• Build a quadratic from a prescribed sum and product — Exercise 2.2 item 2
• Given one zero and the coefficients, find the other, using the sum or the product
• Given the zeroes, write down the quadratic without expanding anything
• Find an unknown coefficient from a stated condition on the sum or product of the zeroes
• Explain why two different quadratics can share the same pair of zeroes

Examples worth working on the board

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

• First specimen: 2x² − 8x + 6 (§2.3, pp. 18–19). The section splits the middle term, aiming for two terms whose product matches 12x². Verified: the split is −6x and −2x, since (−6)(−2) = 12 and (−6) + (−2) = −8; grouping gives 2x(x − 3) − 2(x − 3), then (2x − 2)(x − 3), then 2(x − 1)(x − 3). Zeroes 1 and 3. Verified against the coefficients: sum 4, and −b/a is −(−8)/2 = 4; product 3, and c/a is 6/2 = 3.
• Second specimen: 3x² + 5x − 2 (§2.3, p. 19). Verified: the split is 6x and −x, since (6)(−1) = −6 = 3 × (−2) and 6 − 1 = 5; grouping gives 3x(x + 2) − 1(x + 2), then (3x − 1)(x + 2). Zeroes 1/3 and −2. Verified: sum 1/3 − 2 = −5/3, and −b/a is −5/3; product (1/3)(−2) = −2/3, and c/a is −2/3. Worth showing because the leading coefficient is not 1 and one zero is a fraction, so nothing cancels by accident.
• The derivation (§2.3, p. 19). Let the two zeroes be α and β. Then the two linear expressions x − α and x − β are factors, so ax² + bx + c must equal k(x − α)(x − β) for some constant k. Verified by expanding: k(x − α)(x − β) = k[x² − (α + β)x + αβ] = kx² − k(α + β)x + kαβ. Matching the three coefficients: a = k, b = −k(α + β), and c = kαβ. Substituting k = a into the second gives b = −a(α + β), hence α + β = −b/a; substituting into the third gives c = aαβ, hence αβ = c/a. Note: the constant k turns out to be the leading coefficient itself, and that identification is the step students skip.
• The two relations, as the section presents them (§2.3, pp. 19–20). The sum of the zeroes is minus the coefficient of x over the coefficient of x²; the product of the zeroes is the constant term over the coefficient of x². Show both as ratios with the words in place, since the chapter prints them that way and examinations quote them that way.
• Example 2: x² + 7x + 10 (§2.3, p. 20). Factors (x + 2)(x + 5), zeroes −2 and −5. Verified: sum −7, and −b/a is −7/1 = −7; product 10, and c/a is 10/1 = 10. The sign trap is that both zeroes are negative while the product is positive.
• Example 3: x² − 3 (§2.3, p. 20). The section uses the difference-of-squares identity to write it as a product of two linear factors, giving zeroes √3 and −√3. Verified: sum 0, and the coefficient of x is 0, so −b/a is 0; product −3, and c/a is −3/1 = −3. This is the example that shows the relations do not need whole-number zeroes, and it is also the case where the missing middle term is the evidence.
• Example 4: building from sum −3 and product 2 (§2.3, p. 21). The relations give −b/a = −3 and c/a = 2. Verified: choosing a = 1 forces b = 3 and c = 2, so x² + 3x + 2 fits; and its zeroes are −1 and −2, whose sum is −3 and product is 2, so the construction checks out. The section then notes that every other quadratic meeting the same two conditions is a real multiple of this one. Verified as the reason: multiplying all three coefficients by the same non-zero number leaves both ratios unchanged.
• Exercise 2.2 item 1 (p. 23), six quadratics to factorise and check: x² − 2x − 8; 4s² − 4s + 1; 6x² − 3 − 7x; 4u² + 8u; t² − 15; 3x² − x − 4. Two features worth planning for: the third is printed with its terms out of order, so the coefficient of the first power is −7 and not −3; and the second is a perfect square, so its two zeroes coincide at 1/2, which is where this topic meets Case (ii) of §2.2.
• Exercise 2.2 item 2 (p. 23), six sum-and-product pairs to build from, in the printed order: 1/4 and −1; √2 and 1/3; 0 and √5; 1 and 1; −1/4 and 1/4; 4 and 1. Note that the fractions and roots here are printed stacked inside the question and are easy to mis-transcribe from a text extraction — these are read off the page image.

Figures to have open

• A stacked-comparison panel: the expansion of k(x − α)(x − β) directly above ax² + bx + c, with a line joining each pair of matching coefficients. Not in the book as a figure; this is an added diagram and it carries the whole argument.
• A two-column card showing sum against −b/a and product against c/a, in the ratio form the chapter prints. Standard schematic.
• A fan of curves k(x² + 3x + 2) for several values of k, all crossing the horizontal axis at the same two points. Not in the book; it makes the family in section 12 visible and links this topic back to §2.2.
• 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. 18–21 — the two opening specimens, the general derivation with its Greek letter footnote, and Examples 2, 3 and 4.
• Exercise 2.2 items 1 and 2, p. 23.
• Backward pointer inside the same chapter: the question this section answers was posed at §2.1, p. 11, immediately after the linear case was settled.
• The chapter's §2.4 "Summary", p. 23, keeps the two relations as its fifth point.