NCERT Class 10 Maths Exercise 2.1 Solution – Polynomials

Open the questions in the official book: NCERT Class 10 Maths Chapter 2 (PDF) or read it on the NCERT website.

Video Solution – Exercise 2.1

Watch the Exercise 2.1 solution on YouTube

Step-by-Step Solutions – Exercise 2.1

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Question 1

Question 1 (i)

Find the number of zeroes of $p(x)$ from the graph.

Solution:

1. $$\text{Zeroes = where the }x\text{-axis meets the graph}$$

   Look at the first graph. This straight line is above the x-axis and never touches it.

2. $$\text{Number of zeroes}=0$$

   So it has no zero. The number of zeroes is 0.

Final answer$$\text{Number of zeroes}=0$$

Shortcut / Trick:

$$\begin{array}{l}\text{If the line is parallel to the x-axis}\\ \text{It never meets it = 0 zeroes}\end{array}$$

Remember, if a line is parallel to the x-axis, it never meets it. So the number of zeroes is 0.

Question 1 (ii)

Find the number of zeroes of $p(x)$ from the graph.

Solution:

1. $$\text{Zeroes = where the }x\text{-axis meets the graph}$$

   The second graph cuts the x-axis at only one place, on the right side.

2. $$\text{Number of zeroes}=1$$

   So it has only one zero. The number of zeroes is 1.

Final answer$$\text{Number of zeroes}=1$$

Question 1 (iii)

Find the number of zeroes of $p(x)$ from the graph.

Solution:

1. $$\text{Zeroes = where the }x\text{-axis meets the graph}$$

   In the third graph, the curve first cuts the x-axis on the left. Then it goes up, comes down, and cuts it two more times.

2. $$\text{Number of zeroes}=3$$

   It cuts the x-axis three times in total, so the number of zeroes is 3.

Final answer$$\text{Number of zeroes}=3$$

Question 1 (iv)

Find the number of zeroes of $p(x)$ from the graph.

Solution:

1. $$\text{Zeroes = where the }x\text{-axis meets the graph}$$

   The fourth graph is a parabola that goes below the x-axis, so it cuts the x-axis at two places.

2. $$\text{Number of zeroes}=2$$

   So the number of zeroes is 2.

Final answer$$\text{Number of zeroes}=2$$

Shortcut / Trick:

$$\begin{array}{l}\text{A quadratic polynomial has at most 2 zeroes}\\ \text{Degree n -> max n zeroes}\end{array}$$

Remember this trick. A polynomial of degree n has at most n zeroes. A quadratic has at most 2.

Question 1 (v)

Find the number of zeroes of $p(x)$ from the graph.

Solution:

1. $$\text{Zeroes = where the }x\text{-axis meets the graph}$$

   In the fifth graph, the curve starts from the left and cuts the x-axis one, two, three and four times.

2. $$\text{Number of zeroes}=4$$

   So the number of zeroes is 4.

Final answer$$\text{Number of zeroes}=4$$

Question 1 (vi)

Find the number of zeroes of $p(x)$ from the graph.

Solution:

1. $$\text{Zeroes = where the }x\text{-axis meets the graph}$$

   In the last graph, the curve cuts the x-axis on the left and touches the x-axis at two places in the middle.

2. $$\text{Number of zeroes}=3$$

   So it meets the x-axis three times in total. The number of zeroes is 3.

Final answer$$\text{Number of zeroes}=3$$

Key Takeaways – Exercise 2.1

• A zero of a polynomial p(x) is the x-value where the graph of p(x) crosses or touches the x-axis.
• The number of zeroes of p(x) equals the number of points where its graph intersects the x-axis.
• A graph that does not touch the x-axis at all has zero zeroes (e.g., Q1 part i).
• A polynomial of degree n can have at most n zeroes, which is visible from its graph.
• Exercise 2.1 has only 1 question with 6 graph-based parts – all answers come from reading the graph carefully.

CBSE Board Exam Tips

• In CBSE board exams, a 1-mark MCQ or VSA question is often asked on the number of zeroes from a graph – always count x-axis intersection points carefully, including points where the curve just touches (tangent) the x-axis.
• If the graph touches the x-axis at a point but does not cross it, that point still counts as a zero (with even multiplicity) – do not skip it.
• Remember: a straight line (linear polynomial) can have at most 1 zero, a parabola (quadratic) at most 2, and a cubic curve at most 3 – use this to cross-check your answer from the graph.
• Exercise 2.1 concepts are often tested indirectly in 2-mark questions where you are given the degree and asked the maximum possible zeroes, so make sure you understand the link between degree and number of zeroes.

Practice MCQs – Exercise 2.1

Frequently Asked Questions

How many questions are there in NCERT Class 10 Maths Exercise 2.1?

Exercise 2.1 of Class 10 Maths Chapter 2 (Polynomials) has only 1 question, but it contains 6 parts – (i) to (vi). Each part shows a different graph of a polynomial and asks you to find the number of zeroes of p(x) from that graph.

What is the main concept tested in Exercise 2.1 of Class 10 Maths?

Exercise 2.1 tests the geometrical meaning of the zeroes of a polynomial. The key idea is that the number of zeroes of a polynomial p(x) is equal to the number of times its graph intersects (crosses or touches) the x-axis. No calculations are needed – you just read the graph.

Is Exercise 2.1 of Class 10 Maths important for the CBSE board exam?

Yes, it is important. The concept of finding zeroes from a graph is frequently asked as a 1-mark MCQ or a very short answer question in CBSE board exams. It is also the foundation for understanding the relationship between zeroes and coefficients covered in the rest of Chapter 2.

How do you find the number of zeroes of a polynomial from its graph?

Simply count the number of points where the graph of p(x) meets or cuts the x-axis. Each such point is a zero of the polynomial. If the graph does not touch the x-axis at all, the polynomial has zero zeroes.

What does it mean if the graph of p(x) touches the x-axis but does not cross it?

If the graph touches the x-axis at a point without crossing it, that point is still counted as a zero of the polynomial. This typically happens when the zero has an even multiplicity (for example, a repeated root). So you must count such touching points as zeroes too.

Can a polynomial have more zeroes than its degree?

No, a polynomial of degree n can have at most n zeroes. For example, a quadratic polynomial (degree 2) can have at most 2 zeroes, and a cubic polynomial (degree 3) can have at most 3 zeroes. This is why the graph of a quadratic can cross the x-axis at most 2 times.

Learn More: Polynomials

• Factorization of Polynomials Worksheet
• Long Division of Polynomials Calculator
• Operations with Polynomials: Complete Guide

NCERT Class 10 Maths Solutions – All Exercises

Source: Questions from the NCERT Mathematics Textbook for Class 10, Chapter 2. Solutions written by IrfanEdu.