CBSE · Class 10 · Mathematics · Chapter 2

Chapter 2
Polynomials

Complete chapter resources for CBSE Class 10 Maths — topic breakdown, key formulas, sample questions, previous year board questions, and instant AI question paper generation.

4Topics
4–6Board marks
8Sample questions
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Key Formulas — Chapter 2
  • General polynomial: p(x) = aₙxⁿ + ... + a₁x + a₀
  • Zero of p(x): p(k) = 0 ⟹ k is a zero
  • Sum of zeroes (quadratic): α + β = −b / a
  • Product of zeroes (quadratic): αβ = c / a
  • Division algorithm: p(x) = g(x) · q(x) + r(x)
  • Quadratic from zeroes: x² − (α+β)x + αβ = 0
Chapter Overview

What this chapter covers

A polynomial is an algebraic expression of the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the coefficients are real numbers and the exponents are non-negative integers. Chapter 2 focuses on polynomials up to degree 3 — linear, quadratic, and cubic — and introduces the concept of zeroes, which are the values of x for which p(x) = 0.

A key geometric insight covered in this chapter is that the number of zeroes of a polynomial equals the number of times its graph crosses (or touches) the x-axis. The relationship between zeroes and coefficients — captured by Vieta's formulas — allows students to find the sum and product of zeroes directly from the standard coefficients a, b, c without solving the equation. For a cubic, there are three such relations involving α + β + γ, αβ + βγ + γα, and αβγ.

The Division Algorithm for Polynomials (p(x) = g(x) · q(x) + r(x)) is the algebraic counterpart of the Euclidean algorithm for integers. Board questions apply it to find the remaining zeroes of a cubic when one zero is already known, or to determine whether a polynomial is a factor of another. This tool bridges Chapter 2 with the factorisation methods used in Chapter 4 (Quadratic Equations).

Topics

What's inside Chapter 2

As per NCERT Class 10 Mathematics (CBSE syllabus)

Topic 1
Polynomials and Their Degrees
Definition of polynomials in one variable. Degree, coefficients, constant term. Classification as linear (degree 1), quadratic (degree 2), and cubic (degree 3). Identifying zeroes geometrically from the graph of y = p(x).
Topic 2
Zeroes of a Polynomial
Definition of a zero: p(k) = 0. Geometric meaning — x-intercepts of the graph. A polynomial of degree n has at most n zeroes. Verification by substitution. Linear polynomial has exactly one zero.
Topic 3
Relationship Between Zeroes and Coefficients
Vieta's formulas for quadratic (α + β = −b/a, αβ = c/a) and cubic (α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a) polynomials. Forming a polynomial when its zeroes are known.
Topic 4
Division Algorithm for Polynomials
Statement: p(x) = g(x) · q(x) + r(x), where degree r(x) < degree g(x). Long division of polynomials. Using the algorithm to find quotient and remainder, and to determine the remaining zeroes of a cubic polynomial.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Class 9 · Ch 2 · Polynomials
Degree, coefficients, basic factorisation, Remainder Theorem
Class 9 · Ch 1 · Number Systems
Real numbers as coefficients; irrational zeroes on the number line
Chapter 2 Poly-
nomials
Leads to
Ch 4 · Quadratic Equations
Zeroes of a quadratic become roots of the equation ax² + bx + c = 0
Class 11 · Algebra
Polynomial functions, factor theorem, complex roots, rational root theorem
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 10 board papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Objective 1 1–2 Number of zeroes from graph, or verify a zero by substitution
Very Short Answer 2 1 Find sum/product of zeroes, or form a quadratic from given zeroes
Short Answer 3 1 Find all zeroes of a cubic or apply the division algorithm
Long Answer 4–5 0–1 Division algorithm to find quotient and all zeroes; multi-part problem
Total (approximate) 4–6 3–4 Weightage varies across paper sets and years
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 10 Maths Chapter 2. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
The graph of y = p(x) for a polynomial p(x) cuts the x-axis at exactly 3 points. The number of zeroes of p(x) is: (a) 1 (b) 2 (c) 3 (d) 0
Q2 Easy 2 marks Short Answer
Find a quadratic polynomial whose zeroes are 3 and −2. Write it in standard form.
Q3 Medium 2 marks Short Answer
If α and β are the zeroes of the polynomial 2x² − 7x + 3, find the value of α² + β².
Q4 Medium 3 marks Short Answer
Find the zeroes of the quadratic polynomial 6x² − 3 − 7x and verify the relationship between zeroes and coefficients.
Q5 Medium 3 marks Short Answer
On dividing x³ − 3x² + x + 2 by a polynomial g(x), the quotient and remainder are x − 2 and −2x + 4 respectively. Find g(x).
Q6 Hard 4 marks Long Answer
If two zeroes of the polynomial x⁴ − 6x³ − 26x² + 138x − 35 are 2 ± √3, find the other two zeroes.
Q7 Hard 4 marks Long Answer
Apply the division algorithm to find all zeroes of the polynomial 2x⁴ − 3x³ − 3x² + 6x − 2, given that two of its zeroes are √2 and −√2.
Q8 Hard 5 marks Case-Based
A rectangular park has its length (in metres) represented by the polynomial p(x) = x² + 5x + 6 and its breadth by q(x) = x + 2. (i) Find the zeroes of q(x) and verify using the coefficient relation. (ii) Use the division algorithm to divide p(x) by q(x) and find the quotient. (iii) Interpret what the quotient polynomial represents in the context of the park's dimensions.
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Previous Year Questions

From CBSE board examinations

Actual questions from past Class 10 Maths board papers — Polynomials chapter.

Board 20232 marks
Find the zeroes of the polynomial p(x) = x² − 5 and verify the relationship between zeroes and coefficients. (All India 2023)
Board 20222 marks
If one zero of the quadratic polynomial 2x² + kx − 15 is −3, find the value of k and the other zero. (Delhi 2022)
Board 20203 marks
Find all the zeroes of the polynomial 2x³ + x² − 6x − 3, if two of its zeroes are −√3 and √3. (CBSE 2020)
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FAQ

Questions teachers ask

How many marks does Polynomials carry in the CBSE Class 10 board exam? +
Typically 4–6 marks across 2–3 questions — one 1-mark MCQ on zeroes or the relationship between zeroes and coefficients, one 2-mark short answer on the division algorithm or forming a polynomial, and occasionally a 3-mark question. The chapter has appeared in nearly every CBSE Class 10 Maths board paper over the last five years.
What is a zero of a polynomial and how is it different from the root of an equation? +
A zero of a polynomial p(x) is a value k such that p(k) = 0. When we write p(x) = 0 and solve it, the solutions are called roots of the equation. In practice the terms are interchangeable for Class 10 — both refer to the x-values where the polynomial equals zero and where the graph of y = p(x) crosses the x-axis.
What are the most common board questions from the Polynomials chapter? +
The three most frequent question types are: (1) Given the zeroes of a quadratic, find the polynomial using α + β and αβ relations — typically 2 marks. (2) Verify whether a given value is a zero of a polynomial — 1 mark. (3) Apply the division algorithm to find the remaining zeroes of a cubic or find the quotient and remainder — 3–4 marks. These cover the majority of marks available from this chapter.
How do the sum and product of zeroes formulas work for quadratic and cubic polynomials? +
For a quadratic ax² + bx + c: sum of zeroes α + β = −b/a and product αβ = c/a. For a cubic ax³ + bx² + cx + d: sum α + β + γ = −b/a, sum of products taken two at a time αβ + βγ + γα = c/a, and product αβγ = −d/a. These are Vieta's formulas and are directly tested in CBSE board questions.
How do I generate a custom question paper for Polynomials using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 10 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.