CBSE · Class 9 · Mathematics · Chapter 2

Chapter 2
Polynomials

Complete chapter resources for CBSE Class 9 Maths — topic breakdown, key concepts, sample questions, previous year board questions, and instant AI question paper generation.

4Topics
4–6Board marks
8Sample questions
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Key Concepts — Chapter 2
  • Remainder Theorem: p(x) ÷ (x − a) → remainder = p(a)
  • Factor Theorem: (x − a) is a factor iff p(a) = 0
  • Identity 1: (x + y)² = x² + 2xy + y²
  • Identity 2: (x + y)(x − y) = x² − y²
  • Sum of cubes: x³ + y³ = (x+y)(x²−xy+y²)
  • Difference of cubes: x³ − y³ = (x−y)(x²+xy+y²)
Chapter Overview

What this chapter covers

A polynomial in one variable x is an expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the exponents are non-negative integers and the coefficients aᵢ are real numbers. Chapter 2 begins by classifying polynomials by their degree (linear, quadratic, cubic) and identifying the zeroes — values of x for which p(x) = 0. For a linear polynomial ax + b, there is exactly one zero: x = −b/a.

The two central theorems of this chapter are the Remainder Theorem — which states that dividing a polynomial p(x) by (x − a) gives remainder p(a) — and the Factor Theorem, a special case stating that (x − a) is a factor of p(x) if and only if p(a) = 0. These theorems allow students to check factors and evaluate remainders without performing long polynomial division.

The chapter also establishes seven standard algebraic identities — including expansions of (x + y)², (x − y)², (x + y)³, (x − y)³, and the sum/difference of cubes formulas — which are used to factorise expressions and evaluate numerical expressions such as (99)³ or (103)² without a calculator. These identities are frequently tested in CBSE periodic tests and annual exams.

Topics

What's inside Chapter 2

As per NCERT Class 9 Mathematics (CBSE syllabus)

Topic 1
Polynomials in One Variable
Definition of a polynomial, degree, coefficients, and types — constant, linear, quadratic, cubic. Identifying zeroes of a polynomial and understanding that a polynomial of degree n has at most n zeroes.
Topic 2
Remainder Theorem
When a polynomial p(x) is divided by a linear polynomial (x − a), the remainder is p(a). Used to quickly find remainders without long division and to verify or discover zeroes of polynomials.
Topic 3
Factor Theorem & Factorisation
(x − a) is a factor of p(x) if and only if p(a) = 0. Applied to factorise polynomials of degree 2 and 3 — including the method of splitting the middle term and grouping.
Topic 4
Algebraic Identities
Seven standard identities: (x±y)², (x+a)(x+b), (x+y+z)², (x±y)³, and the sum/difference of cubes. Used to expand expressions, factorise polynomials, and evaluate numerical powers efficiently.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Ch 1 · Number Systems
Real number coefficients and irrational number arithmetic
Class 8 · Algebra
Expansion of algebraic expressions, factorisation basics
Chapter 2 Polynomials
Leads to
Class 10 Ch 2 · Polynomials
Relationship between zeroes and coefficients of quadratics
Class 10 Ch 4 · Quadratic Equations
Factorisation of quadratic polynomials to find roots
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 9 school and board-style papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Objective 1 1–2 Degree, number of zeroes, or remainder identification
Very Short Answer 2 1 Find zero of a polynomial or apply Remainder Theorem
Short Answer 3 1 Factor Theorem application or factorisation of a cubic
Long Answer / Identity Application 4–5 0–1 Multi-step identity expansion or numerical evaluation
Total (approximate) 4–6 3–5 Weightage varies across school papers and periodic tests
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 9 Maths Chapter 2. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
The degree of the polynomial 4x³ − 2x² + 7x − 5 is: (a) 1 (b) 2 (c) 3 (d) 4
Q2 Easy 2 marks Short Answer
Find the zero of the polynomial p(x) = 3x − 9. Verify your answer by substituting back into p(x).
Q3 Medium 2 marks Short Answer
Using the Remainder Theorem, find the remainder when p(x) = x³ − 3x² + 4x − 2 is divided by (x − 2).
Q4 Medium 3 marks Short Answer
Check whether (x + 1) is a factor of p(x) = x³ + x² + x + 1. If yes, factorise p(x) completely.
Q5 Medium 3 marks Short Answer
If p(x) = x² − 5x + k and p(2) = 0, find the value of k. Then factorise p(x) completely.
Q6 Hard 4 marks Short Answer
Factorise the polynomial p(x) = 2x³ − 3x² − 11x + 6 using the Factor Theorem. Show all steps including the trial of rational zeroes.
Q7 Hard 4 marks Word Problem
Without multiplying directly, evaluate (102)³ using the identity (a + b)³ = a³ + 3a²b + 3ab² + b³. Show each step of your working clearly.
Q8 Hard 5 marks Case-Based
If x + y + z = 9 and xy + yz + zx = 26, find the value of x² + y² + z². Also, if xyz = 24, use the identity for (x + y + z)³ to find x³ + y³ + z³ − 3xyz. (i) Find x² + y² + z². (ii) Find x³ + y³ + z³ − 3xyz. (iii) Hence find x³ + y³ + z³.
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Previous Year Questions

From CBSE board examinations

Actual questions from past Class 9 Maths papers — Polynomials chapter.

Board 20222 marks
Find the value of k if (x − 1) is a factor of 4x³ + 3x² − 4x + k. (CBSE Class 9 Annual 2022)
Board 20233 marks
Factorise: x³ − 23x² + 142x − 120 using the Factor Theorem. (CBSE Class 9 Annual 2023)
Board 20203 marks
If a + b + c = 5 and ab + bc + ca = 10, find a² + b² + c². Also find a³ + b³ + c³ − 3abc. (CBSE Class 9 SA-II 2020)
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FAQ

Questions teachers ask

How many marks does Polynomials carry in the CBSE Class 9 Mathematics exam? +
Polynomials typically carries 4–6 marks in CBSE Class 9 annual and periodic assessments. Expect one 1-mark MCQ testing degree or coefficient identification, one 2-mark question on zeroes or the Remainder Theorem, and one 3-mark question applying the Factor Theorem or expanding algebraic identities. School-level papers often vary, but the chapter is consistently tested every year.
What is the Remainder Theorem and how is it tested in Class 9 exams? +
The Remainder Theorem states that if a polynomial p(x) is divided by (x − a), the remainder equals p(a). In exams, questions typically give p(x) and ask for the remainder when divided by a linear binomial, or give the remainder and ask for an unknown coefficient. These are usually 2-mark questions and require substitution only — no long division needed.
What is the difference between the Remainder Theorem and the Factor Theorem? +
The Factor Theorem is a special case of the Remainder Theorem. If p(a) = 0, then (x − a) is a factor of p(x) — that's the Factor Theorem. If p(a) = r (where r ≠ 0), r is the remainder — that's the Remainder Theorem. Board questions on the Factor Theorem ask students to show that a given binomial is a factor of p(x), or to find the value of k if (x − a) is a factor.
Which algebraic identities from Chapter 2 are most important for CBSE Class 9 exams? +
The five high-priority identities are: (x + y)² = x² + 2xy + y², (x − y)² = x² − 2xy + y², (x + y)(x − y) = x² − y², (x + a)(x + b) = x² + (a+b)x + ab, and the sum/difference of cubes: x³ + y³ = (x + y)(x² − xy + y²) and x³ − y³ = (x − y)(x² + xy + y²). Questions routinely ask for expansions, factorisations, or numerical evaluations using these identities.
How do I generate a custom question paper for Polynomials using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 9 Mathematics, select Chapter 2 (Polynomials), set your preferred question-type mix and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.