These NCERT solutions for class 10 maths chapter 2 cover both exercises in the Polynomials chapter. Exercise 2.1 has one question and Exercise 2.2 has two questions, each with six subparts. This page tells you what each exercise tests and which method it wants, so you land on the right solution page the first time.
The chapter runs on one idea: the zeroes of a polynomial are the points where its graph crosses the x-axis, and the same zeroes are tied to the polynomial’s coefficients. Hold onto that link and every question in both exercises follows the same pattern.
| Exercise | What you get |
|---|---|
| Exercise 2.1 | Solved |
| Exercise 2.2 | Solved |
Exercise 2.1 and Exercise 2.2: what each one asks
The chapter holds two small exercises. The table below lists each one, how many questions it carries, and the method it tests; use it to find the exercise your homework question sits in.
Exercise 2.1 — the chapter’s only graph-reading question. It shows several graphs of \(y = p(x)\) and asks for the number of zeroes in each case. The method is purely visual: count the points where each curve cuts or touches the x-axis.
Exercise 2.2 — the algebra test, in two questions. Question 1 asks you to find the zeroes of six quadratics and verify the relationship with the coefficients; the set deliberately mixes a perfect square, a difference of squares and a binomial that only needs a common factor taken out.
Question 2 works in reverse: it hands you the sum and the product of the zeroes and asks you to construct the polynomial.
The method map: factorise, compare, or read the graph
Three techniques carry the whole chapter, and each exercise drills one of them.
- Graph reading — used by Exercise 2.1. A zero of \(p(x)\) is the x-coordinate of a point where the graph of \(y = p(x)\) meets the x-axis. A polynomial of degree \(n\) has at most \(n\) zeroes. A parabola can cut the axis at two distinct points, touch it at one point, or miss it entirely.
- Factorisation — used by Exercise 2.2 Question 1. Split the middle term of a quadratic, use the difference of squares \(a^2 – b^2 = (a – b)(a + b)\) for forms such as \(x^2 – 3\) and \(t^2 – 15\), or take out a common factor for a binomial like \(4u^2 + 8u\).
- Coefficient comparison — used by Exercise 2.2 Questions 1 and 2. Once the zeroes are known, check their sum and product against the coefficients; when sum and product are given, reverse the same relationship to build the polynomial.
The key results connect zeroes to coefficients.
| Polynomial | Sum of zeroes | Product of zeroes |
|---|---|---|
| Quadratic \(ax^2 + bx + c\) | \(\alpha + \beta = -\frac{b}{a}\) | \(\alpha\beta = \frac{c}{a}\) |
| Cubic \(ax^3 + bx^2 + cx + d\) | \(\alpha + \beta + \gamma = -\frac{b}{a}\) | \(\alpha\beta\gamma = -\frac{d}{a}\) |
For a cubic, there is also the sum of the products taken two at a time: \(\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}\).
When a question gives you the sum \(S\) and product \(P\) of the zeroes, any polynomial of the form \(k(x^2 – Sx + P)\) with \(k \neq 0\) fits — the answer is not unique.
Common mistakes that cost marks in Polynomials
- Sign slips in \(-\frac{b}{a}\). When \(b\) itself is negative, the formula gives a positive sum. Dropping the minus sign in front of \(b\) flips the answer.
- Cubic product sign. For a cubic, the product of zeroes is \(-\frac{d}{a}\), not \(\frac{d}{a}\). Losing the minus gives the wrong constant term.
- One touch counts as one zero. A graph that touches the x-axis at one point gives two coincident zeroes — the polynomial has one distinct zero, not two.
- Forgetting the constant \(k\). When you build a polynomial from a given sum and product, the answer is any multiple \(k(x^2 – Sx + P)\), \(k \neq 0\). Writing only one version loses generality.
- Wrong tool on a difference of squares. Splitting the middle term fails on forms like \(t^2 – 15\); use \(a^2 – b^2\) instead.
- Forgotten condition \(a \neq 0\). Every formula divides by the coefficient of the highest power. If that coefficient is zero, the expression is not a quadratic at all.
How to use this page before your homework
- If your question shows graphs of \(y = p(x)\) and asks for the number of zeroes, open Exercise 2.1 solutions.
- If it gives a quadratic and asks you to find the zeroes and verify the relationship, open Exercise 2.2 Question 1 solutions.
- If it gives the sum and product of zeroes and asks for a polynomial, open Exercise 2.2 Question 2 solutions.
This page only routes you; the step-by-step working lives on the linked exercise pages. For the rest of the course, browse the Class 10 Maths solutions, move on to pair of linear equations in two variables, or revise real numbers. You can also check any formula against the official NCERT Polynomials chapter PDF.
For wider revision, see the Class 10 hub or the CBSE notes index.
| Exercise | What you get |
|---|---|
| Exercise 2.1 | Solved |
| Exercise 2.2 | Solved |
Frequently asked questions about Class 10 Maths Chapter 2
How many exercises are there in Class 10 Maths Chapter 2 Polynomials?
Two: Exercise 2.1 with one question and Exercise 2.2 with two questions, each having six subparts.
Which exercise has questions on finding the zeroes of a quadratic polynomial and verifying the relationship?
Exercise 2.2 Question 1. It gives six quadratics to factorise, find the zeroes of, and verify against the coefficients.
What is the relationship between the zeroes and the coefficients of a quadratic polynomial?
For \(ax^2 + bx + c\), the sum of the zeroes is \(-\frac{b}{a}\) and the product is \(\frac{c}{a}\).
How do I find a quadratic polynomial when the sum and product of its zeroes are given?
Use \(k(x^2 – Sx + P)\), where \(S\) is the sum, \(P\) is the product and \(k\) is any non-zero constant — the answer is not unique.
What does the graph of a quadratic polynomial look like?
A parabola. It opens upwards when \(a \gt 0\) and downwards when \(a \lt 0\).
Reference: NCERT textbooks (CBSE).