NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials
Chapter 2 of Class 9 Maths introduces students to polynomials, one of the most important building blocks of algebra. The chapter begins with the definition of a polynomial in one variable and moves on to classify polynomials based on the number of terms (monomial, binomial, trinomial) and degree (linear, quadratic, cubic). Students learn to find the zeros of a polynomial, apply the Remainder Theorem and Factor Theorem, and use standard algebraic identities to factorise expressions quickly without long division.
NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials break down every exercise question into simple, exam-ready steps that follow the CBSE format. Since polynomials form the base for quadratic equations, arithmetic progressions, and calculus in higher classes, a strong grasp of this chapter pays off well beyond the Class 9 exam. Concepts like the Factor Theorem and the identities for expanding (a+b)³ or (x+a)(x+b)(x+c) are asked almost every year in board-pattern papers, making this one of the highest-weightage chapters in the algebra unit. These NCERT Solutions for Class 9 help students identify the fastest method for each type of question, whether it involves factorisation, division, or direct substitution, so they can save time during exams while still showing complete, accurate working.
Find the PDF of NCERT Solutions for Class 9 Maths Chapter 2: Polynomials
Students preparing for exams or revising at home can download the free PDF of NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials. It includes fully worked solutions for every question in the chapter, covering zeros of a polynomial, the Remainder and Factor Theorems, and algebraic identities, making it a handy offline resource for quick revision.
Important Points of Chapter 2 – Polynomials
Topic | Key Point |
|---|---|
Polynomial | An algebraic expression with terms having non-negative integer exponents |
Degree of Polynomial | Highest power of the variable in the polynomial |
Types by Terms | Monomial (1 term), Binomial (2 terms), Trinomial (3 terms) |
Types by Degree | Linear (degree 1), Quadratic (degree 2), Cubic (degree 3) |
Zero of a Polynomial | Value of the variable that makes the polynomial equal to zero |
Remainder Theorem | Gives the remainder when a polynomial is divided by (x − a) |
Factor Theorem | Tells whether (x − a) is a factor of the polynomial |
Factorisation | Expressing a polynomial as a product of its factors |
Algebraic Identities | Standard formulas used to expand or factorise expressions quickly |
Important Formulas of Chapter 2 – Polynomials
Formula | Description |
|---|---|
(a + b)² = a² + 2ab + b² | Square of a sum |
(a − b)² = a² − 2ab + b² | Square of a difference |
a² − b² = (a + b)(a − b) | Difference of squares |
(x + a)(x + b) = x² + (a+b)x + ab | Product of two binomials |
(a + b)³ = a³ + 3a²b + 3ab² + b³ | Cube of a sum |
(a − b)³ = a³ − 3a²b + 3ab² − b³ | Cube of a difference |
a³ + b³ = (a + b)(a² − ab + b²) | Sum of cubes |
a³ − b³ = (a − b)(a² + ab + b²) | Difference of cubes |
a³ + b³ + c³ − 3abc = (a+b+c)(a²+b²+c²−ab−bc−ca) | Sum of three cubes identity |
(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx | Square of a trinomial |
Important Concepts of Polynomials for Exams
Understanding Zeros of a Polynomial: A zero of a polynomial p(x) is a value of x for which p(x) = 0. This is not the same as the polynomial equalling zero everywhere; it refers to specific input values. Exam questions frequently ask students to verify whether a given number is a zero of a polynomial or to find the zero of a linear polynomial, so this concept needs to be practised until it becomes second nature.
Remainder Theorem in Depth: The Remainder Theorem states that when a polynomial p(x) is divided by (x − a), the remainder is p(a). This shortcut avoids the lengthy process of long division and is a favourite among exam setters because it tests both conceptual clarity and calculation accuracy. Students should practise substituting values carefully, especially with negative numbers and fractions.
Applying the Factor Theorem: Closely related to the Remainder Theorem, the Factor Theorem says that (x − a) is a factor of p(x) if and only if p(a) = 0. This is used extensively in factorisation problems where students must first test possible factors before applying the identity-based methods. Many higher-order thinking (HOTS) questions in exams combine the Factor Theorem with algebraic identities to test deeper understanding.
Factorisation Using Identities: A major portion of the chapter, and consequently a major portion of the exam weightage, involves factorising polynomials using standard identities. Recognising which identity applies to a given expression, whether it is a perfect square, a difference of squares, or a sum/difference of cubes, is a skill that comes only with regular practice. Misidentifying the correct identity is one of the most common mistakes students make under exam pressure.
Expanding Cubes and Trinomials: Questions involving (a+b)³, (a−b)³, and the expansion of a trinomial squared are common in both short-answer and long-answer sections. Students should memorise these identities along with their proofs, since some papers ask for a derivation rather than direct application.
Word Problems and Applications: Some exercises apply polynomial concepts to real-life or geometric contexts, such as finding the volume of a cuboid expressed as a product of its dimensions. These questions test whether students can translate a written problem into a polynomial expression and then simplify or factorise it correctly.
A thorough understanding of zeros, the two theorems, and all standard identities will make Chapter 2 one of the more scoring units in the Class 9 Maths syllabus, provided students revise regularly using the NCERT exercises and the solutions PDF.
FAQs on NCERT Solutions for Class 9 Maths Chapter 2 Polynomials
A polynomial is an algebraic expression made up of variables, constants, and whole-number exponents combined using addition, subtraction, or multiplication. In Class 9 Maths Chapter 2, students learn how to identify whether an expression is a polynomial by checking that the powers of variables are non-negative integers. Expressions containing negative exponents, fractional powers, or variables in the denominator are not polynomials. The chapter also explains important terms such as coefficients, variables, constants, and degrees of polynomials. NCERT Solutions guide students through numerous examples that make identification easier and reduce confusion during examinations. By practising different types of expressions, students develop the ability to recognise polynomials quickly and solve related questions accurately in school tests and competitive examinations.
The degree of a polynomial is one of the most important concepts in this chapter because it determines the classification of the polynomial. NCERT Solutions explain that the degree is the highest exponent of the variable present in the expression. Students also learn to identify constant, linear, quadratic, and cubic polynomials based on their degrees. Every exercise includes step-by-step explanations that help students avoid common mistakes while determining the degree. The solutions use simple examples before introducing more challenging problems, allowing learners to build confidence gradually. Regular practice with these solved questions improves conceptual clarity and prepares students for exam questions that require identifying, comparing, and classifying different types of polynomials correctly.
The Remainder Theorem is an important concept introduced in the Polynomials chapter to simplify division problems involving polynomials. It states that when a polynomial is divided by a linear polynomial of the form (x − a), the remainder is obtained by substituting the value of 'a' into the polynomial. Instead of performing lengthy polynomial division every time, students can quickly calculate the remainder using this theorem. NCERT Solutions explain the theorem with clear examples and solved exercises that help students understand both the logic and application. Mastering this concept improves problem-solving speed and accuracy. It also lays the foundation for more advanced algebraic topics studied in higher classes, making it an essential part of Class 9 Mathematics.
To perform well in the Polynomials chapter, students should begin by understanding the basic terminology, including variables, coefficients, terms, degree, and types of polynomials. After learning these concepts, they should practise addition, subtraction, multiplication, and evaluation of polynomial expressions. It is equally important to understand identities and the Remainder Theorem through regular practice. NCERT Solutions provide detailed, step-by-step answers that explain the reasoning behind every calculation, making learning easier and more effective. Students should revise solved examples, attempt all exercise questions independently, and review their mistakes carefully. Consistent practice not only improves calculation skills but also builds confidence in solving application-based questions. With thorough preparation and regular revision, students can score excellent marks in this chapter.