CBSE Class 10 Maths Chapter 2 Polynomials NCERT Solutions
This resource provides detailed NCERT Solutions for Class 10 Mathematics, Chapter 2: Polynomials. It focuses on understanding the concept of zeroes of a polynomial by interpreting their graphical representations. The solutions explain how the number of zeroes corresponds to the number of times the graph of a polynomial intersects the x-axis. This chapter is crucial for building a strong foundation in algebra, and these solutions offer step-by-step guidance to master the graphical method of finding zeroes, aiding students in their exam preparation and conceptual clarity.
Quick info
| Board | CBSE |
|---|---|
| Class | Class 10 |
| Subject | Mathematics |
| Session | 2026 |
| Language | English |
| Type | NCERT Solutions |
| Chapter | Chapter 2 |
Chapter summary
Chapter 2 of the NCERT Class 10 Mathematics textbook introduces Polynomials. These solutions specifically address Exercise 2.1, focusing on finding the number of zeroes of a polynomial p(x) by examining the provided graphs of y = p(x). The core concept is that the number of intersections of the graph with the x-axis directly indicates the number of real zeroes for the polynomial.
Learning outcomes
- Understand the relationship between the graph of a polynomial and its zeroes.
- Determine the number of zeroes of a polynomial from its graphical representation.
- Interpret the geometrical meaning of zeroes of a polynomial.
- Solve problems involving finding the number of zeroes from given graphs.
Topics covered
Paper topics
- Polynomials
- Zeroes of a polynomial
- Graphical representation of polynomials
- Number of zeroes
- Relationship between zeroes and graph
- Exercise 2.1
Important topics
- Graphical interpretation of zeroes
- Counting x-axis intersections
- Understanding p(x) = 0 geometrically
- Zeroes of polynomials
PDF preview
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Questions and Solutions
Exercise 2.1
Question 1
The graphs of y = p(x) are given in the following figures, for some polynomials p(x). Find the number of zeroes of p(x), in each case. (i)
(ii)
(iii)
(iv)
(v)
(vi)
The number of zeroes of a polynomial p(x) is determined by the number of points where its graph, , intersects the x-axis. Each intersection point on the x-axis represents a real value of x for which .
- Case (i): The graph intersects the x-axis at 3 distinct points. Therefore, the polynomial p(x) has 3 zeroes.
- Case (ii): The graph intersects the x-axis at exactly 1 point. Therefore, the polynomial p(x) has 1 zero.
- Case (iii): The graph intersects the x-axis at 2 distinct points. Therefore, the polynomial p(x) has 2 zeroes.
- Case (iv): The graph intersects the x-axis at exactly 1 point. Therefore, the polynomial p(x) has 1 zero.
- Case (v): The graph intersects the x-axis at 4 distinct points. Therefore, the polynomial p(x) has 4 zeroes.
- Case (vi): The graph intersects the x-axis at 3 distinct points. Therefore, the polynomial p(x) has 3 zeroes.
Common mistakes
- Confusing x-axis intersections with y-axis intersections.
- Miscounting the number of intersection points.
- Assuming a graph must intersect the x-axis to have zeroes.
- Not understanding that touching the x-axis counts as an intersection.
Revision tips
- Focus on the definition of a zero: a value of x for which p(x) = 0.
- Practice identifying intersection points with the x-axis on various graphs.
- Remember that each distinct intersection point represents a unique real zero.
- Review the provided graphs and solutions to reinforce the graphical interpretation.
Practice MCQs
Q1. What does the number of zeroes of a polynomial p(x) represent geometrically?
Explanation: The number of zeroes of a polynomial p(x) is equal to the number of points where the graph of y = p(x) intersects the x-axis.
Q2. If a graph of y = p(x) touches the x-axis at one point, how many zeroes does the polynomial have?
Explanation: Touching the x-axis at a single point means the polynomial has exactly one real zero at that point.
Q3. For a polynomial p(x), if its graph does not intersect the x-axis at all, what can be concluded about its real zeroes?
Explanation: If the graph of y = p(x) never crosses or touches the x-axis, the polynomial has no real values of x for which p(x) = 0.
Q4. In the context of graphical representation, what does p(x) = 0 signify?
Explanation: The equation p(x) = 0 represents the condition for finding the zeroes of the polynomial, which correspond to the x-intercepts of its graph.
Frequently asked questions
What is the main concept covered in these NCERT Solutions for Class 10 Maths Chapter 2?
These solutions focus on understanding and finding the number of zeroes of a polynomial by analyzing the graphical representation of y = p(x), specifically by counting the points where the graph intersects the x-axis.
How does the graph of y = p(x) help in finding the number of zeroes?
The number of times the graph of y = p(x) intersects or touches the x-axis corresponds directly to the number of real zeroes of the polynomial p(x).
What if the graph of y = p(x) does not intersect the x-axis?
If the graph does not intersect the x-axis at any point, it means the polynomial p(x) has no real zeroes.
Are these solutions useful for exam preparation?
Yes, these solutions provide clear explanations and graphical interpretations, which are essential for mastering the concept of zeroes of polynomials and performing well in exams.
What is the significance of the x-axis in these graphical interpretations?
The x-axis represents the values of x for which the polynomial's value, p(x), is equal to zero. Therefore, the points of intersection on the x-axis are the zeroes of the polynomial.
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