CBSE • Class 10 • Mathematics • Chapter 2 (Polynomials) • Exercise 2.1

Exercise 2.1: Polynomials — NCERT Solutions

Geometric meaning of zeros — reading zeros of a polynomial off its graph.

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What this exercise covers

Linear graphsQuadratic graphsReading zeros

Step-by-step solutions — Exercise 2.1

4 solved questions • Each solution includes a Socratic hint, full working and a common-mistake callout • Last reviewed 2026-09-03

Exercise 2.1 Q1 • 1 marks

The graph of a polynomial y=p(x)y = p(x) cuts the x-axis at exactly two distinct points. How many zeros does the polynomial have?
Hint (Socratic — try this first)▾
What does each point where the graph meets the x-axis represent?
Step-by-step solution▾

The zeros of a polynomial are precisely the x-coordinates of the points where the graph of y=p(x)y = p(x) intersects the x-axis.

Here the graph cuts the x-axis at two distinct points.

Therefore the polynomial has 2 zeros\textbf{2 zeros}.

Common mistake:
Counting points where the graph crosses the y-axis (there is only ever one such point) instead of the x-axis.
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Exercise 2.1 Q2 • 1 marks

A graph of y=p(x)y = p(x) touches the x-axis at one point and does not cross it anywhere else. What can you say about the number of zeros of p(x)p(x)?
Hint (Socratic — try this first)▾
When a curve just touches the axis without crossing, how many times does it meet the axis?
Step-by-step solution▾

When a graph touches the x-axis at a point (rather than crossing it), the graph still meets the x-axis at that single point.

So there is exactly 1 point of intersection with the x-axis.

Hence p(x)p(x) has 1 zero\textbf{1 zero} (this is a repeated zero, but geometrically only one distinct point of contact).

Common mistake:
Saying the polynomial has no zeros because the graph does not 'cross' the axis; touching the axis still counts as meeting it.
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Exercise 2.1 Q3 • 1 marks

The graph of a certain polynomial does not intersect the x-axis at any point. How many real zeros does it have?
Hint (Socratic — try this first)▾
If a graph never meets the x-axis, at how many x-values is p(x)=0p(x) = 0?
Step-by-step solution▾

A real zero occurs where y=p(x)=0y = p(x) = 0, i.e. where the graph meets the x-axis.

Since the graph never meets the x-axis, there is no value of xx for which p(x)=0p(x) = 0.

Therefore the polynomial has 0 (no) real zeros\textbf{0 (no) real zeros}.

Common mistake:
Confusing 'no real zeros' with 'not a polynomial'; a polynomial can exist and simply have no real zeros.
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Exercise 2.1 Q4 • 1 marks

The graph of y=p(x)y = p(x) is a straight line that meets the x-axis at exactly one point. What is the degree of the polynomial and how many zeros does it have?
Hint (Socratic — try this first)▾
What is the shape of the graph of a linear polynomial?
Step-by-step solution▾

The graph of a linear polynomial ax+bax + b (with a≠0a \neq 0) is a straight line.

A non-horizontal straight line meets the x-axis at exactly one point.

  • Degree of the polynomial =1= \textbf{1} (linear).
  • Number of zeros =1= \textbf{1}.
Common mistake:
Assuming a straight line must be quadratic; a line is a degree-1 polynomial with exactly one zero.
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How to approach Exercise 2.1

  1. Re-read the chapter summary first. Open Polynomials and refresh the key concepts: Zeros of a polynomial, Sum and product of zeros, Quadratic polynomials, Cubic polynomials.
  2. Try each problem yourself before opening the solution. Ten focused minutes per problem usually beats reading two finished solutions.
  3. Use Guru AI for the questions you get stuck on. The Socratic AI tutor walks you through it with hints instead of dictating the answer.
  4. Mark the questions you got wrong and revisit them after 24 hours — the spacing is what locks the method into long-term memory.

All exercises in Polynomials

  1. Exercise 2.1 — Geometric meaning of zeros — reading zeros of a polynomial off its graph.
  2. Exercise 2.2 — Relationship between zeros and coefficients of a quadratic polynomial.

Exercise 2.2 →

Relationship between zeros and coefficients of a quadratic polynomial.

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