Class 10 Maths Chapter 2 Polynomials NCERT Solutions – Ex 2.1, 2.2, 2.3, 2.4 | narrowit.in

Welcome to narrowit.in’s NCERT Solutions for Class 10 Maths Chapter 2 – Polynomials. This page covers complete step‑by‑step solutions for Exercise 2.1, Exercise 2.2, Exercise 2.3, and Exercise 2.4 (Optional). Topics include the geometrical meaning of zeroes, the relationship between zeroes and coefficients, the division algorithm for polynomials, and finding all zeroes of a polynomial when some are given. All solutions are written by narrowit.in in simple language – no login required, no app download, completely free. Official NCERT textbooks are available at https://ncert.nic.in.

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Class 10 Maths Chapter 2 – Polynomials is a fundamental chapter that bridges the gap between simple algebra and advanced calculus. Polynomials are expressions containing variables, constants, and exponents, and they appear everywhere in mathematics – from physics equations to economic models. In the CBSE curriculum, this chapter introduces students to the algebraic structure of polynomials, their graphical representations, and the powerful relationships between their zeroes and coefficients. Mastering this chapter is essential for scoring well in board examinations and for building a strong foundation for higher classes, where polynomial functions become central to calculus and coordinate geometry.

This chapter revisits and deepens the concept of polynomials that students first encountered in Class 9. In Class 10, the focus shifts from mere definition to a deeper analysis: understanding how polynomials behave geometrically, how their zeroes relate to their coefficients, and how to divide one polynomial by another using a systematic algorithm.

Class 10 Maths Ch1 NCERT Solutions –> Click Here

Exercise 2.1 – Geometrical Meaning of Zeroes

Exercise 2.1 is based on the geometrical interpretation of the zeroes of a polynomial. Students are given graphs of polynomial functions and are asked to find the number of zeroes by observing where the graph intersects the x‑axis. A zero of a polynomial p(x) is the x‑coordinate of a point where the graph of y = p(x) cuts or touches the x‑axis. A linear polynomial (degree 1) has exactly one zero – its graph is a straight line that meets the x‑axis at exactly one point. A quadratic polynomial (degree 2) has a parabolic graph which can intersect the x‑axis at zero, one, or two points – giving zero, one, or two zeroes respectively. A cubic polynomial (degree 3) can have one, two, or three zeroes. The key rule to remember is that a polynomial of degree n can have at most n zeroes, though it may have fewer. This exercise builds a strong visual intuition for what zeroes represent, making abstract algebra much more concrete. It is a scoring topic in board exams as questions are straightforward once the concept of graph‑axis intersection is clear.

Ex 2.1 PDF:

Exercise 2.2 – Relationship between Zeroes and Coefficients

Exercise 2.2 introduces one of the most powerful tools in polynomial algebra – the relationship between the zeroes of a polynomial and its coefficients. For a quadratic polynomial ax² + bx + c with zeroes α and β, the sum of zeroes is given by α + β = –b/a, and the product of zeroes is αβ = c/a. For a cubic polynomial ax³ + bx² + cx + d with zeroes α, β, γ, the relationships extend to: α + β + γ = –b/a, αβ + βγ + γα = c/a, and αβγ = –d/a. This exercise requires students to find zeroes of given quadratic polynomials by factorisation, and then verify that the sum and product of the zeroes match the coefficients using these formulas. It also includes questions where students are given the sum and product of zeroes and are asked to construct the polynomial. This is one of the most important concepts in the chapter as it appears in almost every board exam paper, often in application‑based problems.

Exercise 2.3 – Division Algorithm for Polynomials

Exercise 2.3 covers the Division Algorithm for Polynomials, which states that for any two polynomials p(x) and g(x) with g(x) ≠ 0, there exist unique polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where either r(x) = 0 or the degree of r(x) is less than the degree of g(x). Here, p(x) is the dividend, g(x) is the divisor, q(x) is the quotient, and r(x) is the remainder. This algorithm is analogous to the division algorithm for integers and is used to check whether one polynomial is a factor of another. Questions in this exercise require students to divide polynomials step by step, find quotients and remainders, and verify the result using the division formula. It also includes word problems where students must find a divisor or quotient given the other components. This exercise is essential for understanding how polynomials behave under division and is the gateway to more advanced topics like factorisation and partial fractions in higher classes.

Exercise 2.4 (Optional) – Finding All Zeroes

Exercise 2.4 is an optional exercise, but it is extremely important for students aiming for high scores. It builds directly on the division algorithm and teaches students how to find all zeroes of a polynomial when some zeroes are already known. The method involves identifying the known zeroes, forming the corresponding factors, dividing the polynomial to find the quotient, and then solving the quotient to find the remaining zeroes. This exercise also reinforces the concept of the factor theorem: if (x – a) is a factor of a polynomial p(x), then a is a zero of p(x). Students are often given a polynomial with some zeroes and asked to find the others, testing their ability to apply division and factorisation together. The questions in this exercise are more challenging and require careful step‑by‑step working, but they are frequently asked in board exams and competitive tests. Mastering this exercise gives students a significant advantage.

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