NCERT Solutions for Class 10 Maths Chapter 2: Polynomials

Chapter 2 Polynomials is the second chapter of CBSE Class 10 Mathematics and belongs to the Algebra unit, which carries 20 marks in total across Chapters 2 through 5. Polynomials typically carries 3 to 4 marks in the CBSE Class 10 board exam 2027. This chapter builds on the polynomial concepts introduced in Class 9 and focuses on the relationship between zeros and coefficients of quadratic polynomials, graphical interpretation of zeros, and forming polynomials from given zeros.

What Are Polynomials?

A polynomial is an algebraic expression made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, 2x^3 - 5x^2 + 3x - 7 is a polynomial in x. The highest power of the variable is called the degree of the polynomial. A polynomial of degree 1 is called linear (e.g., 3x + 2), degree 2 is called quadratic (e.g., x^2 - 5x + 6), and degree 3 is called cubic (e.g., x^3 - 4x).

The general form of a quadratic polynomial is ax^2 + bx + c, where a is not equal to 0. In Class 10, most of the work focuses on quadratic polynomials because the relationship between zeros and coefficients is especially elegant and frequently tested in board exams.

Zeros of a Polynomial

A zero of a polynomial p(x) is a value of x for which p(x) = 0. For example, if p(x) = x^2 - 5x + 6, then p(2) = 4 - 10 + 6 = 0 and p(3) = 9 - 15 + 6 = 0. So 2 and 3 are the zeros of this polynomial. Finding zeros means solving the equation p(x) = 0.

For quadratic polynomials, zeros are found by factorisation. The standard method is to split the middle term. For example, to factorise x^2 - 2x - 8, we look for two numbers that multiply to give -8 and add to give -2. These numbers are -4 and 2. So x^2 - 2x - 8 = (x - 4)(x + 2), and the zeros are 4 and -2.

An important property is that a polynomial of degree n has at most n zeros. A linear polynomial has exactly 1 zero, a quadratic has at most 2 zeros, a cubic has at most 3 zeros, and so on. Some quadratic polynomials like x^2 + 1 have no real zeros because x^2 + 1 is always positive for real values of x.

Relationship Between Zeros and Coefficients

This is the most important concept in Chapter 2 and the one tested most frequently. For a quadratic polynomial ax^2 + bx + c with zeros alpha and beta:

Sum of zeros: alpha + beta = -b/a Product of zeros: alpha x beta = c/a

These formulas come from the factorisation ax^2 + bx + c = a(x - alpha)(x - beta). Expanding the right side and comparing coefficients gives the relationships.

For example, for the polynomial 3x^2 - 5x + 2: a = 3, b = -5, c = 2. The sum of zeros = -(-5)/3 = 5/3 and the product = 2/3. Students can verify by factoring: 3x^2 - 5x + 2 = (3x - 2)(x - 1), zeros are 2/3 and 1. Sum = 2/3 + 1 = 5/3 and product = 2/3 x 1 = 2/3.

Conversely, if the sum of zeros is S and the product is P, then the quadratic polynomial is x^2 - Sx + P or any non-zero scalar multiple k(x^2 - Sx + P). This allows us to form a polynomial from its zeros.

For cubic polynomials ax^3 + bx^2 + cx + d with zeros alpha, beta, gamma: alpha + beta + gamma = -b/a alpha.beta + beta.gamma + gamma.alpha = c/a alpha.beta.gamma = -d/a

These cubic relationships are tested in state board exams but not in CBSE 2026-27 since the related exercises have been removed.

Graphical Meaning of Zeros

The zeros of a polynomial p(x) are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis. This is a simple but powerful concept that appears as MCQs in every board exam.

For a linear polynomial ax + b, the graph is a straight line that intersects the x-axis at one point (-b/a, 0). So it has exactly one zero.

For a quadratic polynomial, the graph is a parabola. If it intersects the x-axis at two distinct points, the polynomial has 2 distinct zeros. If it touches the x-axis at one point, both zeros are equal (the parabola just touches). If it does not intersect the x-axis at all, the polynomial has no real zeros.

The shape of the parabola depends on the sign of the leading coefficient. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward.

Division Algorithm (For State Boards)

The Division Algorithm for polynomials states that for any two polynomials p(x) and g(x) where g(x) is not zero, there exist polynomials q(x) (quotient) and r(x) (remainder) such that p(x) = g(x) x q(x) + r(x), where the degree of r(x) is less than the degree of g(x), or r(x) is zero.

This is analogous to the division algorithm for integers (Euclid's Division Lemma from Chapter 1). The process of polynomial long division is used to find the quotient and remainder.

This topic has been removed from the CBSE 2026-27 syllabus but remains relevant for state board students in UP, Bihar, MP, Rajasthan, and Haryana. Padhaao provides solutions for the older exercises (2.3 and 2.4) for these students.

How This Chapter Is Tested in Board Exams

Polynomials questions in the CBSE board exam typically follow these patterns. One MCQ worth 1 mark tests the graphical meaning of zeros (how many zeros from a given graph) or the sum/product relationship (find the sum of zeros given a polynomial). One short answer question worth 2 to 3 marks asks students to find the zeros of a given quadratic polynomial and verify the relationship between zeros and coefficients. Alternatively, students may be asked to form a quadratic polynomial given the sum and product of zeros.

In recent years, the 'reciprocal zeros' pattern has been very popular: If one zero of kx^2 + bx + c is the reciprocal of the other, find k. Since the product of reciprocal zeros is 1, c/a = 1 gives the answer directly. Questions asking for expressions like 1/alpha + 1/beta or alpha^2 + beta^2 in terms of coefficients are also common.

Assertion-Reason questions on polynomials test conceptual clarity about degree, zeros, and coefficient relationships. Case study questions typically involve a real-world scenario where a quadratic polynomial models a physical situation like the path of a ball or the area of a field.

Tips for Scoring Full Marks

To score all 3 to 4 marks from this chapter, students should focus on four things. First, memorise the sum and product formulas perfectly, including the negative sign in the sum formula (-b/a, not b/a). Second, practice factorisation of all 6 polynomials in Exercise 2.2 Q1 until it becomes automatic. Third, always write the verification step when finding zeros, since examiners award separate marks for it. Fourth, solve at least 10 previous year questions to recognise the common patterns like reciprocal zeros, alpha^2 + beta^2, and 1/alpha + 1/beta.

Common mistakes to avoid include forgetting the negative sign in the sum formula, confusing sum and product formulas, not rearranging polynomials into standard form before applying formulas, and counting turning points instead of x-axis intersections in graph questions.

Connection to Other Chapters

Polynomials connects directly to Chapter 4 (Quadratic Equations). The zeros of a quadratic polynomial ax^2 + bx + c are exactly the roots of the quadratic equation ax^2 + bx + c = 0. The sum and product of roots formulas in Chapter 4 are the same as the sum and product of zeros in Chapter 2. Students who master polynomials will find quadratic equations much easier.

The concept of factorisation used to find zeros also appears in Chapter 3 (Pair of Linear Equations) when solving systems by substitution or elimination. Understanding graphs of polynomials helps with the graphical method of solving linear equations.

For students preparing for competitive exams like NTSE and Olympiads, the polynomial relationships extend to higher-degree polynomials and connect to advanced topics like symmetric functions of roots and polynomial identities.