Polynomials — Class 10 Mathematics (CBSE) explained

What you’ll learn

1. 1

   Lesson 1: From Degree to Graphical Zeroes

   6 min
   • Recall the Degree of a Polynomial
   • Recognise Linear Polynomials
   • Quadratic and Cubic: Names and General Forms
   • What Does the Value of a Polynomial Mean?
   • A Zero Makes the Polynomial's Value Zero
   • The Zero of a Linear Polynomial
   • A Linear Zero Is an x-Coordinate
   • From a Quadratic Value Table to Its Zeroes
2. 2

   Lesson 2: Counting Zeroes and Discovering Relationships

   5 min
   • Two Distinct Zeroes or One Zero
   • No Zeroes and the Meaning of At Most
   • A Cubic with Three Zeroes
   • A Cubic Need Not Have Three Distinct Zeroes
   • Example 1: Count the Intersections
   • Factorise the First Quadratic
   • Compare the Zeroes with the Coefficients
3. 3

   Lesson 3: Zeroes and Coefficients: Quadratic Applications

   5 min
   • A Second Quadratic Confirms the Pattern
   • Why the Quadratic Relationships Hold
   • Example 2: Find and Verify the Zeroes
   • Example 3: Zeroes Involving Square Roots
   • Example 4: Work Backwards from Sum and Product
   • A Cubic: Sum and Product of Three ZeroesNot assessed
   • The Third Relationship: Products Two at a TimeNot assessed
4. 4

   Lesson 4: Cubic Relationships, Enrichment and Practice

   5 min
   • Organise the Three Cubic RelationshipsNot assessed
   • Optional Example 5: Verify the First Two ZeroesNot assessed
   • Optional Example 5: Verify the Fractional ZeroNot assessed
   • Optional Example 5: Verify the Sum RelationshipNot assessed
   • Optional Example 5: Verify Pair Products and the ProductNot assessed
   • Exercise 2.2: Two Directions of Practice
   • Chapter Recap: Read, Relate and Practise

Key concepts

Linear classification

A polynomial of degree one is called a linear polynomial.

Quadratic and cubic forms

A polynomial of degree two is called a quadratic polynomial. The name comes from quadrate, meaning square.

Definition of a zero

A zero is an input number that makes the output zero. The input itself need not be zero.

Two quadratic intersection cases

Two separate markers give two distinct input values. One coincident marker gives one input value.

Quadratic upper bound

A quadratic can have two distinct zeroes, two equal zeroes described as one zero, or no zero.

Degree and maximum zero count

Degree provides a maximum, not an exact count.

Exam focus

Board exam

CBSE assesses this chapter in the 2026-27 year-end board examination.

Syllabus

POLYNOMIALS: 1. Zeros of a polynomial. 2. Relationship between zeros and coefficients of quadratic polynomials. Learning outcome: find the zeros of polynomial graphically and algebraically and verifying the relation between zeros and coefficients of quadratic polynomials.

Not assessed

These parts go beyond the 2026-27 syllabus and are taught for understanding only: A Cubic: Sum and Product of Three Zeroes, The Third Relationship: Products Two at a Time, Organise the Three Cubic Relationships, Optional Example 5: Verify the First Two Zeroes, Optional Example 5: Verify the Fractional Zero, Optional Example 5: Verify the Sum Relationship and Optional Example 5: Verify Pair Products and the Product.

Exam note

The relationship between zeroes and coefficients is in the syllabus for QUADRATIC polynomials only; the textbook's cubic relationships (and the cubic worked example) are outside the 2026-27 syllabus.

Paper pattern

Mathematics Standard: 80 marks, 3 hours, 38 questions. Section A: 18 MCQs + 2 assertion-reason (1 mark each); B: 5 VSA (2 marks); C: 6 SA (3 marks); D: 4 LA (5 marks); E: 3 case-study questions (4 marks). No calculators.

Question style

About 50% of board questions are competency-focused.

Practice in the app

67 board-pattern questions with marking schemes: 21 multiple choice (1 mark), 10 assertion–reason (1 mark), 15 very short answer (2 marks), 10 short answer (3 marks), 4 long answer (5 marks) and 7 case-based (4 marks).

Sample questions with marking points

Board-pattern questions from the AI Lessons practice set for this chapter. The marking points show what an examiner looks for; full model answers are in the app.

1. Very short answer2 marks

   For p(x) = x² - 3x - 4, a student writes p(-1) = -1 + 3 - 4 = -2. Identify the error and evaluate p(-1) correctly.

   Marking points

   • Identify the incorrect squaring of -1 and state that (-1)² = 1.1
   • Correctly substitute and obtain p(-1) = 0.1

2. Short answer3 marks

   Using 3x² + 5x - 2 = (3x - 1)(x + 2), a student reports the zeroes as 1/3 and 2. Correct the error and verify both relationships between the zeroes and coefficients.

   Marking points

   • Sets 3x - 1 = 0 and x + 2 = 0.0.5
   • Obtains the correct zeroes 1/3 and -2.0.5
   • Calculates the sum as -5/3 and compares it with -b/a = -5/3.1
   • Calculates the product as -2/3 and compares it with c/a = -2/3.1

3. Long answer5 marks

   For p(x) = 2x² - 8x + 6, use a valid middle-term split to find the zeroes and verify their sum and product against the coefficients. Without plotting the graph, state all its x-axis intersection points and justify why there cannot be another such point.

   Marking points

   • Chooses -6x and -2x and checks their sum -8x and product 12x².0.5
   • Completes the factorisation through grouping to obtain 2(x - 1)(x - 3).1
   • Obtains the zeroes 1 and 3.0.5
   • Verifies the sum using 1 + 3 = 4 = -(-8)/2.1
   • Verifies the product using 1 × 3 = 3 = 6/2.1
   • States both x-axis intersection points: (1, 0) and (3, 0).0.5
   • Justifies that no additional x-axis intersection is possible because a quadratic has at most two distinct real zeroes.0.5

Frequently asked questions

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