Relationship Between Zeroes and Coefficients of a Polynomial (Class 10)

In one line: for a quadratic polynomial ax² + bx + c with zeroes α and β, the sum of the zeroes is α + β = −b/a and the product of the zeroes is αβ = c/a.

This is one of the most useful ideas in the whole Polynomials chapter, because it lets you know the sum and product of the zeroes straight from the coefficients — without solving the polynomial at all. If you can read off a, b and c, you already know two important facts about the zeroes.

For a cubic polynomial ax³ + bx² + cx + d with zeroes α, β, γ, the same style of relationship holds: α + β + γ = −b/a, αβ + βγ + γα = c/a, and αβγ = −d/a. This guide covers Section 2.3 of your NCERT chapter — the quadratic case in full, the cubic case in brief — and it builds directly on the meaning of a zero of a polynomial.

▶️ Watch the 30-second reel: Find the sum of zeroes! 🤔

What are the zeroes and coefficients? (quick recap)

A zero of a polynomial p(x) is a value of x, say k, for which p(k) = 0. A quadratic polynomial can have at most two zeroes; a cubic can have at most three.

The coefficients are the fixed numbers multiplying each power of x. In the standard quadratic form ax² + bx + c:

• a = coefficient of x² (must not be 0),
• b = coefficient of x,
• c = constant term.

For example, in 2x² − 8x + 6 we have a = 2, b = −8 and c = 6. Being able to name these correctly — with their signs — is the whole skill, so it is worth being slow and careful here. A quick refresher on classifying polynomials lives in Types of Polynomials — linear, quadratic, cubic.

The relationship for a quadratic polynomial

For a quadratic polynomial ax² + bx + c (with a ≠ 0) whose zeroes are α and β:

• Sum of zeroes: α + β = −b/a = −(coefficient of x) ÷ (coefficient of x²)
• Product of zeroes: αβ = c/a = (constant term) ÷ (coefficient of x²)

Where these come from (the intuition). If α and β are the zeroes, then (x − α) and (x − β) are factors, so we can write

ax² + bx + c = a(x − α)(x − β) = a[x² − (α + β)x + αβ] = ax² − a(α + β)x + aαβ.

Comparing the matching terms on both sides:

• comparing the x terms: b = −a(α + β), which rearranges to α + β = −b/a,
• comparing the constant terms: c = aαβ, which rearranges to αβ = c/a.

Conditions: this applies to any quadratic with a ≠ 0, and it works whether the zeroes are whole numbers, fractions, or irrational — you do not even need the zeroes to be “nice.”

Worked example 1 — a first check

Take p(x) = 2x² − 8x + 6. Splitting the middle term, 2x² − 8x + 6 = 2(x − 1)(x − 3), so the zeroes are 1 and 3. Here a = 2, b = −8, c = 6.

Both sides agree, so the relationship checks out.

▶️ Watch the 30-second reel: Product of zeroes in seconds! 🤔

Worked example 2 — verifying the relationship (exam-style)

Find the zeroes of x² + 7x + 10 and verify the relationship between the zeroes and the coefficients.

Factorise: x² + 7x + 10 = (x + 2)(x + 5), so the zeroes are −2 and −5. Here a = 1, b = 7, c = 10.

• Sum from zeroes: (−2) + (−5) = −7; from coefficients: −b/a = −7/1 = −7 ✓
• Product from zeroes: (−2) × (−5) = 10; from coefficients: c/a = 10/1 = 10 ✓

The relationship is verified. This “find, then verify” format is exactly how the question is asked in the board exam, and presenting both the zero-side and coefficient-side clearly is what earns full method marks — see how CBSE maths answers are marked.

Worked example 3 — irrational zeroes still work

Verify the relationship for x² − 3.

Using a² − b² = (a − b)(a + b), we get x² − 3 = (x − √3)(x + √3), so the zeroes are √3 and −√3. Here a = 1, b = 0, c = −3.

• Sum: √3 + (−√3) = 0; −b/a = −0/1 = 0 ✓
• Product: (√3)(−√3) = −3; c/a = −3/1 = −3 ✓

Even with irrational zeroes, the coefficients still fix the sum and product perfectly. Working a mixed set of these — whole-number, fractional and irrational zeroes together — is the fastest way to make the pattern automatic; the Topic Practice sets on examfront group them for exactly this, and the Practice Companion flags the step where you slip.

▶️ Watch the 30-second reel: One zero is given—find the other!

Watch the sign: sum uses a minus

The single most common error in this topic is dropping the minus sign in the sum. Read the rule carefully:

• Sum = −b/a → there is a minus in front.
• Product = c/a → no minus.

So for x² − 5x + 6, the coefficient b is −5, and the sum of zeroes is −b/a = −(−5)/1 = +5 (the two minuses cancel). The zeroes are 2 and 3, and indeed 2 + 3 = 5. If you had written “−5”, that would be wrong. A reliable memory line: “Sum flips the sign of b; product just copies c.”

▶️ Watch the 30-second reel: Find k without solving!

The relationship for a cubic polynomial

The same idea extends one step further. For a cubic polynomial ax³ + bx² + cx + d (with a ≠ 0) whose zeroes are α, β, γ:

• Sum of the zeroes: α + β + γ = −b/a
• Sum of the products taken two at a time: αβ + βγ + γα = c/a
• Product of all the zeroes: αβγ = −d/a

Notice the pattern of signs: the sum uses −b/a and the full product uses −d/a (both with a minus), while the middle relation uses c/a (no minus).

Worked example — a clean cubic check

Take p(x) = x³ − 6x² + 11x − 6 = (x − 1)(x − 2)(x − 3), so the zeroes are 1, 2, 3. Here a = 1, b = −6, c = 11, d = −6.

All three relationships hold. In the CBSE exam the cubic relationships are usually asked as a verification (you are given the zeroes to check), so recognising the three formulas and applying them cleanly is enough. The harder, mixed cubic problems — where you must use one relationship to unlock another — are worked through step by step inside examfront, where the Mistake Identification tool pinpoints exactly which sign or term tripped you up.

▶️ Watch the 30-second reel: Can you find αβ + βγ + γα? 🤔

Where this is heading next

Once you can go from a polynomial to the sum and product of its zeroes, the natural next skill is the reverse: building a quadratic polynomial when you are told the sum and product of its zeroes (using x² − (sum)x + product). That reverse skill is a favourite board-exam question and is covered in its own guide — practising both directions together, and tracking your accuracy with Progress Tracking on examfront, is what turns this into guaranteed marks.

Key Takeaways

• For a quadratic ax² + bx + c with zeroes α, β: α + β = −b/a and αβ = c/a.
• The sum has a minus sign (−b/a); the product does not (c/a).
• You can read off the sum and product of the zeroes straight from the coefficients, without solving.
• The relationship holds for whole-number, fractional and irrational zeroes alike.
• For a cubic ax³ + bx² + cx + d with zeroes α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a.
• To verify, compute the sum/product from the zeroes and from the coefficients and check they match.

Quick Facts

• Sum of zeroes (quadratic): −b/a = −(coefficient of x) ÷ (coefficient of x²).
• Product of zeroes (quadratic): c/a = (constant term) ÷ (coefficient of x²).
• Cubic — sum: α + β + γ = −b/a.
• Cubic — two-at-a-time: αβ + βγ + γα = c/a.
• Cubic — full product: αβγ = −d/a.
• Condition: a ≠ 0 in every case.
• Chapter: Polynomials · Class: 10 · Subject: Maths · Board: CBSE.

Common Mistakes

1. Dropping the minus in the sum. The sum is −b/a, not b/a. For x² − 5x + 6 the sum is −(−5)/1 = 5, not −5. Fix: always write the minus first, then substitute b with its sign.
2. Forgetting to divide by a. With a ≠ 1, both formulas divide by a. For 2x² − 8x + 6 the product is 6/2 = 3, not 6. Fix: the coefficient of x² is always the denominator.
3. Reading coefficients without their signs. In x² + 7x + 10, b = +7; in x² − 7x + 10, b = −7 — these give opposite sums. Fix: copy each coefficient together with the sign in front of it.
4. Using the cubic product without the minus. For a cubic, αβγ = −d/a (with a minus), unlike the quadratic product c/a. Fix: remember “cubic full product carries a minus.”
5. Skipping the two-sided check in “verify” questions. Showing only the zero side (or only the coefficient side) loses marks. Fix: always compute both and state that they match.

FAQ

Q. What is the relationship between the zeroes and coefficients of a polynomial? For a quadratic ax² + bx + c with zeroes α and β, the sum is α + β = −b/a and the product is αβ = c/a. The coefficients fix the sum and product of the zeroes directly.

Q. What is the formula for the sum of zeroes of a quadratic polynomial? Sum of zeroes = −b/a = −(coefficient of x) ÷ (coefficient of x²). The minus sign matters: for x² − 5x + 6 the sum is −(−5)/1 = 5.

Q. What is the product of zeroes of a quadratic polynomial? Product of zeroes = c/a = (constant term) ÷ (coefficient of x²). For x² − 5x + 6 the product is 6/1 = 6, which matches the zeroes 2 and 3.

Q. What is the relationship between zeroes and coefficients of a cubic polynomial? For ax³ + bx² + cx + d with zeroes α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a, and αβγ = −d/a. The full product of the three zeroes carries a minus sign.

Q. How do you verify the relationship between zeroes and coefficients? Find the zeroes by factorising, compute their sum and product, then compute −b/a and c/a from the coefficients. If both pairs match, the relationship is verified.

Related Concepts

• Zeroes of a Polynomial — meaning and how to find: what a zero is and how to compute it — the starting point for this topic.
• Geometrical Meaning of the Zeroes of a Polynomial: how these same zeroes appear as x-axis crossings on a graph.
• Types of Polynomials — linear, quadratic, cubic: the degree-based classification that sets a, b, c, d.
• Quadratic Equations Class 10 — chapter guide: where sum and product of roots return in the very next topic.
• Polynomials Class 10 — how to approach the chapter: the full chapter roadmap and where this section fits.

Ready to make this automatic? Verify a few polynomials, then flip to building them, inside the Topic Practice on examfront — the Practice Companion shows you exactly where a sign slips, and Progress Tracking tells you when you have mastered it. Start on examfront →