Polynomials Class 10: Complete Notes & Solved Examples

Q: What is a zero of a polynomial?

A zero of a polynomial p(x) is a value of x for which p(x) = 0. Geometrically, the zeroes of a polynomial are the x-coordinates of the points where its graph intersects the x-axis.

Q: What is the relationship between zeroes and coefficients of a quadratic polynomial?

For a quadratic polynomial ax² + bx + c with zeroes α and β: the sum of zeroes α + β = -b/a, and the product of zeroes αβ = c/a. These relationships let you form or verify a quadratic polynomial without fully solving it.

Q: What is the relationship between zeroes and coefficients of a cubic polynomial?

For a cubic polynomial ax³ + bx² + cx + d with zeroes α, β and γ: the sum of zeroes α+β+γ = -b/a, the sum of products of zeroes taken two at a time αβ+βγ+γα = c/a, and the product of all three zeroes αβγ = -d/a.

Q: What is the Division Algorithm for polynomials?

If p(x) and g(x) are two polynomials with g(x) ≠ 0, then there exist unique polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where r(x) = 0 or the degree of r(x) is strictly less than the degree of g(x).

Q: How many zeroes can a polynomial of degree n have?

A polynomial of degree n can have at most n zeroes. A linear polynomial has at most 1 zero, a quadratic polynomial has at most 2 zeroes, and a cubic polynomial has at most 3 zeroes.

Q: What are the most common mistakes in Class 10 Polynomials?

Common mistakes include confusing degree with the number of terms, forgetting the negative sign in the sum-of-zeroes formula (-b/a), mixing up the sign in the cubic product-of-zeroes formula (-d/a), and forgetting that the remainder in the Division Algorithm must have a degree strictly less than the divisor's degree.

Polynomials Class 10 complete notes and solved examples

Polynomials Class 10: Complete Notes with Solved Examples (2026 Guide)

📅 Updated Sep 3, 2026 ⏱ 12 min read 📘 CBSE / NCERT Class 10, Chapter 2 ✍️ MyTestSeries Expert Team

Polynomials Class 10 notes — zeroes, degree and division algorithm explained by MyTestSeries

Right after Real Numbers, Class 10 Maths hands you Polynomials — and the jump can feel sudden. Suddenly you're not just finding HCF and LCM of numbers; you're finding the "zeroes" of expressions, connecting them back to coefficients through formulas, and dividing one polynomial by another the way you once divided numbers. The good news: the whole chapter rests on just three ideas — what a zero is, how zeroes relate to coefficients, and how polynomial division works.

This guide lays out every formula in the chapter with a solved example beside it, followed by the specific mistakes that quietly cost marks in exams — sign errors, degree confusion, and skipped verification steps.

4Core Concepts

6Solved Examples

6Common Mistakes

1Quick Formula Table

📋 Table of Contents

What Is a Polynomial? Degree & Types
Zeroes of a Polynomial & Their Geometric Meaning
Zeroes & Coefficients — Quadratic Polynomial
Zeroes & Coefficients — Cubic Polynomial
Division Algorithm for Polynomials
6 Fully Solved Examples
Common Mistakes Students Make
Quick Formula Recap Table
How to Revise This Chapter Effectively
Frequently Asked Questions
Related Test Series & Practice Sets

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1. What Is a Polynomial? Degree & Types

A polynomial in one variable x is an algebraic expression of the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the exponents of x are whole numbers (0, 1, 2, …) and the coefficients aₙ, aₙ₋₁, …, a₀ are real numbers. The degree of a polynomial is the highest power of x present — and it's the degree, not the number of terms, that determines the polynomial's type and its maximum number of zeroes.

Type	Degree	General Form	Example
Linear	1	ax + b (a ≠ 0)	3x − 5
Quadratic	2	ax² + bx + c (a ≠ 0)	2x² − 3x + 1
Cubic	3	ax³ + bx² + cx + d (a ≠ 0)	x³ − 6x² + 11x − 6

💡 Quick check: 5x² + 3x − 7 + x⁵ has degree 5 (from the x⁵ term), even though it's written with the highest power last. Always scan every term before deciding the degree.

2. Zeroes of a Polynomial & Their Geometric Meaning

x = k is a zero of p(x) if and only if p(k) = 0. Geometrically, the zeroes of p(x) are exactly the x-coordinates of the points where the graph y = p(x) crosses or touches the x-axis.

This graph connection is the reason a polynomial's degree caps its number of zeroes: a straight line (linear, degree 1) crosses the x-axis at most once; a parabola (quadratic, degree 2) crosses it at most twice; a cubic curve crosses it at most three times. In general, a polynomial of degree n has at most n zeroes.

⚠️ Watch out: "At most n zeroes" does not mean "always exactly n real zeroes." A quadratic graph that stays entirely above or below the x-axis has zero real zeroes, even though its degree is 2.

3. Zeroes & Coefficients — Quadratic Polynomial

For p(x) = ax² + bx + c with zeroes α and β:
Sum of zeroes: α + β = −b/a
Product of zeroes: αβ = c/a These let you construct a quadratic directly: p(x) = k[x² − (sum)x + (product)], for any non-zero constant k.

This is one of the highest-utility formulas in the chapter — it lets you build or verify a quadratic without ever factorising or using the quadratic formula.

Not sure the sum-product formula has fully clicked? Try it on real questions with the 7-Day Demo Test Series — instant, explained solutions after every attempt.

4. Zeroes & Coefficients — Cubic Polynomial

For p(x) = ax³ + bx² + cx + d with zeroes α, β and γ:
α + β + γ = −b/a
αβ + βγ + γα = c/a
αβγ = −d/a Notice the alternating sign pattern: −b/a, +c/a, −d/a.

💡 Memory trick: The sign alternates starting with negative — sum is −b/a, pairwise-sum is +c/a, product is −d/a. Students who forget this pattern almost always get the product-of-zeroes sign wrong.

5. Division Algorithm for Polynomials

p(x) = g(x) × q(x) + r(x) where r(x) = 0, or degree of r(x) < degree of g(x). p(x) = dividend, g(x) = divisor, q(x) = quotient, r(x) = remainder.

This is the polynomial equivalent of Euclid's Division Lemma from the Real Numbers chapter — divide one polynomial by another, and you get a unique quotient and a remainder whose degree is strictly smaller than the divisor's. It's used to check whether a given expression is a factor of a polynomial (remainder = 0), and to find remaining zeroes once one zero or factor is already known.

6. Six Fully Solved Examples

Example 1 — Finding zeroes and verifying the relationship

Find the zeroes of p(x) = x² − 3x − 4, and verify the relationship between zeroes and coefficients.

Factorising: x² − 4x + x − 4 = x(x−4) + 1(x−4) = (x−4)(x+1). So zeroes are x = 4 and x = −1.
Sum: 4 + (−1) = 3 = −b/a = −(−3)/1 = 3 ✅
Product: 4 × (−1) = −4 = c/a = −4/1 = −4 ✅

Example 2 — Constructing a quadratic from sum and product

Find a quadratic polynomial whose sum and product of zeroes are −3 and 2 respectively.

Using p(x) = k[x² − (sum)x + (product)]:
p(x) = k[x² − (−3)x + 2] = k(x² + 3x + 2). Taking k = 1: p(x) = x² + 3x + 2

Example 3 — Cubic polynomial verification

Verify the relationship between zeroes and coefficients for p(x) = x³ − 6x² + 11x − 6, given its zeroes are 1, 2 and 3.

Here a=1, b=−6, c=11, d=−6.
Sum: 1+2+3 = 6 = −b/a = −(−6)/1 = 6 ✅
Pairwise sum: (1×2)+(2×3)+(3×1) = 2+6+3 = 11 = c/a = 11/1 = 11 ✅
Product: 1×2×3 = 6 = −d/a = −(−6)/1 = 6 ✅

Example 4 — Dividing one polynomial by another

Divide p(x) = x³ − 3x² + 5x − 3 by g(x) = x² − 2, and find the quotient and remainder.

Step 1: x³ ÷ x² = x. Multiply: x(x²−2) = x³−2x. Subtract: (x³−3x²+5x−3) − (x³−2x) = −3x²+7x−3.
Step 2: −3x² ÷ x² = −3. Multiply: −3(x²−2) = −3x²+6. Subtract: (−3x²+7x−3) − (−3x²+6) = 7x−9.
Degree of (7x−9) = 1, which is less than degree of g(x) = 2, so we stop here.
Quotient q(x) = x − 3, Remainder r(x) = 7x − 9
Check: g(x)·q(x)+r(x) = (x²−2)(x−3) + (7x−9) = x³−3x²−2x+6+7x−9 = x³−3x²+5x−3 = p(x) ✅

Example 5 — Zeroes with reciprocal relationships

Find a quadratic polynomial whose zeroes are the reciprocals of the zeroes of x² − 5x + 6.

Zeroes of x² − 5x + 6 are 2 and 3 (since it factors as (x−2)(x−3)).
Reciprocal zeroes: 1/2 and 1/3.
New sum = 1/2 + 1/3 = 5/6, new product = 1/2 × 1/3 = 1/6.
p(x) = k[x² − (5/6)x + 1/6]. Multiply by 6 to clear fractions (k=6): p(x) = 6x² − 5x + 1

Example 6 — Finding a polynomial's unknown coefficient

If one zero of the quadratic polynomial 2x² − 8x + k is 3, find the value of k.

Since x = 3 is a zero, p(3) = 0: 2(3)² − 8(3) + k = 0 → 18 − 24 + k = 0 → −6 + k = 0 → k = 6

7. Common Mistakes Students Make

⚠️ Mistake 1 — Confusing degree with number of terms. A polynomial with 4 terms isn't automatically degree 4 — check the highest exponent of x, not how many terms are written.

⚠️ Mistake 2 — Dropping the negative sign in the sum formula. Students often write α + β = b/a instead of the correct α + β = −b/a. This single sign error cascades into every later step.

⚠️ Mistake 3 — Sign error in the cubic product formula. The product of zeroes of a cubic is −d/a, not d/a. Forgetting the alternating sign pattern (−, +, −) is the single most common error in this section.

⚠️ Mistake 4 — Stopping division too early or too late. Division must continue exactly until the remainder's degree is strictly less than the divisor's degree — not equal to it, and not left one step short.

⚠️ Mistake 5 — Assuming every quadratic has two real zeroes. A quadratic can have two real zeroes, one repeated real zero, or no real zeroes — always match this to what the graph or the given data actually shows.

⚠️ Mistake 6 — Not verifying the final answer. After finding zeroes, always plug the sum and product back into the coefficient relations — this two-line check catches most sign and arithmetic errors before they cost marks.

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8. Quick Formula Recap Table

Concept	Formula
Zero of a polynomial	p(k) = 0
Max zeroes of degree-n polynomial	At most n zeroes
Quadratic — sum of zeroes	α + β = −b/a
Quadratic — product of zeroes	αβ = c/a
Constructing a quadratic	p(x) = k[x² − (sum)x + (product)]
Cubic — sum of zeroes	α+β+γ = −b/a
Cubic — sum of pairwise products	αβ+βγ+γα = c/a
Cubic — product of zeroes	αβγ = −d/a
Division Algorithm	p(x) = g(x)·q(x) + r(x), deg r < deg g

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9. How to Revise This Chapter Effectively

📖 Connect zeroes to the graph before memorising formulas. Picture where a line, parabola, or cubic curve crosses the x-axis — it makes "at most n zeroes" intuitive instead of abstract.

✍️ Write the sign pattern out by hand. For a cubic: −b/a, +c/a, −d/a. Say it out loud a few times until the alternating sign is automatic.

🧮 Practise polynomial long division like arithmetic long division. Example 4 above is the exact template CBSE expects — align terms by descending power and subtract carefully at each step.

🎯 Always verify. After finding zeroes, spend 20 seconds checking sum and product against the coefficients — this habit alone prevents most silly-mistake mark losses.

🔁 Revisit this page before your unit test. Bookmark it, and use the Quick Formula Recap Table (Section 8) as your final five-minute revision before the exam.

📚 Also Useful on MyTestSeries

Real Numbers Class 10 — Formulas, Solved Examples & Common Mistakes (the chapter right before this one)
Class 10 Mathematics Foundation Online Test Series — chapter-wise, topic-wise & full-syllabus mocks
Class 8 Maths: Complete Chapter-wise Formula Sheet
NCERT vs Reference Books — Class 9 Strategy Guide
CBSE vs ICSE vs State Board — Which Is Better for Competitive Exams?
CBSE & ICSE Blog — All Study Guides & Exam Updates

🌐 Trusted External References

NCERT Official Website — download the official Class 10 Maths textbook
CBSE Academic — official syllabus & sample papers
Polynomial — Wikipedia — broader mathematical background
Polynomial Long Division — Wikipedia — extended worked examples

10. Frequently Asked Questions

What is a zero of a polynomial?

A zero of a polynomial p(x) is a value of x for which p(x) = 0. Geometrically, it's the x-coordinate of a point where the graph of the polynomial meets the x-axis.

What is the relationship between zeroes and coefficients of a quadratic polynomial?

For ax² + bx + c with zeroes α and β: α + β = −b/a, and αβ = c/a.

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.

What is the Division Algorithm for polynomials?

For polynomials p(x) and g(x) (g(x) ≠ 0), there exist unique polynomials q(x) and r(x) such that p(x) = g(x)·q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).

How many zeroes can a polynomial of degree n have?

At most n zeroes. A linear polynomial has at most 1, a quadratic at most 2, and a cubic at most 3 — though the actual real-zero count can be fewer.

What are the most common mistakes in Class 10 Polynomials?

Confusing degree with the number of terms, dropping the negative sign in α+β = −b/a, mixing up the sign in the cubic product formula (−d/a), and stopping polynomial division before the remainder's degree is strictly less than the divisor's.

Conclusion: Three Ideas, One Chapter

Polynomials looks like a long chapter on paper, but it's really three connected ideas: what a zero means, how zeroes relate back to coefficients, and how polynomial division works. Once you can move between these three fluently — spotting a zero, deriving a polynomial from its zeroes, or dividing to find a factor — the rest of the exercises become variations on the same six patterns shown above.

Rework the six solved examples without looking at the answers, say the cubic sign pattern out loud until it's automatic, and then test yourself under timed conditions. That's the sequence that actually converts formulas into marks.

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Tags: Polynomials Class 10, Zeroes of a Polynomial, Relationship Between Zeroes and Coefficients, Division Algorithm for Polynomials, CBSE 2026, NCERT Class 10 Maths Chapter 2

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