NCERT Solutions Class 10 Maths Chapter 2 Polynomials Exercise 2.1

NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.1 is a foundational exercise in understanding the fundamental concepts of polynomials. In this exercise of Chapter 2, we will deal with the conceptually important things regarding polynomials, including their categories, forms, and zeroes, explained in detail to help students build a clear understanding of the concept. The material is aligned with the most recent CBSE syllabus and exam standards. Let’s dive into this critical area of study.

1.0Download NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.1 : Free PDF

2.0Introduction to Polynomials

A polynomial is an algebraic expression that has one or more terms, with each term comprising a variable to a non-negative integer power multiplied by a coefficient. The terms are added or subtracted. Polynomials can be classified according to the degree, with the variable's maximum power defining the degree of the polynomial. For example, 4x+1 is a polynomial with degree 1, while 2x²–3y+4 is a polynomial with degree 2. 

Types of Polynomials

1. Linear Polynomial: A polynomial with degree 1 is called a linear polynomial and can be written in general form as ax+b, where $a\ne 0$. Examples of linear polynomials include 2x–3, x + y + 2. 
2. Quadratic Polynomial: Quadratic polynomial is a polynomial with degree 2 in general form written as $a{x}^2+bx+c$, where $a\ne 0$. Examples include, $x^2-3x-4$ and $5x^2-2x+3.$
3. Cubic Polynomial: A polynomial with degree 3 is known as a cubic polynomial. The general form of a cubic polynomial is $a{x}^3+b{x}^2+cx+d$, where $a\ne 0$. Examples of cubic Polynomials includes, $x^3-4x,2x^3-3x^2+x-1$

3.0Exercise 2.1 Overview: Key Concepts

1. Zeroes of Polynomials: The zero of a polynomial is the value of the variable present in that polynomial, which makes it equal to 0. Zeroes of polynomials are also known as the solution of polynomials. For example, give a polynomial, say, p(x) = x² – 3x – 4, substituting x = 2, we will get p(2) = 0, so x = 2 is one of the zeroes of the polynomial. Note that the degree of a polynomial is always equal to the number of zeroes present in a polynomial with one variable. 
2. Geometrical Interpretation of the Zeroes of Polynomials: In this section, we will look at the geometrical meaning of the zeroes of polynomials by analysing their graphs. The zeroes of a polynomial are the x-intercepts of its graph. The number of x-intercepts a polynomial has is equal to the number of its zeroes. For instance, a degree 1 linear polynomial can cross the x-axis at most once, whereas a degree 2 quadratic polynomial can cross it at most twice. In the same vein, a degree 3 cubic polynomial can cross the x-axis at most three times.
3. Zeroes of a Linear Polynomial: Zeroes of a Linear Polynomial of form ax + b, where $a\ne 0$, make a straight line while representing on a graph and the zero of the polynomial is the point of intersection of that line on the x-axis. In other words, for a linear polynomial, the zero is the x-coordinate of the point where the graph intersects the x-axis. 
4. Zeroes of a Quadratic Polynomial: The shape of the graphical representation of a quadratic polynomial say ax² + bx + c, where $a\ne 0$ is a parabola. The zeroes of the quadratic polynomial are the x-coordinates where the graph intersects the x-axis.  A quadratic polynomial has three cases of intersection of zeroes on the graph, which are: 

• Two different zeroes: The graph cuts the x-axis at two points making a parabola. 
• One zero: The graph just touches the x-axis at only one point (two points which coincide) and either downwards or upwards in a parabolic shape. 
• No zeroes: The graph never crosses the x-axis, when there are no real roots present for a quadratic equation. 

5. Zeroes of a Cubic Polynomial: For a cubic polynomial, say ax³ + bx² + cx + d, where $a\ne 0$, the graph of this equation will intersect the x-axis up to three times. The number of zeroes of a cubic polynomial is equal to 3, which are the points where the graph intersects the x-axis. 

Also Read: CBSE Notes On Class 10 Maths Chapter 2 Polynomials

4.0NCERT Class 10 Maths Chapter 2 Exercise 2.1: Detailed Solutions

1. The graphs of y=p(x) are given below, for some polynomials p(x). Find the number of zeros of p(x), in each case.

Solutions:

(i) Graph of y=p(x) does not intersect the x-axis. Hence, polynomial p(x) has no zero.

(ii) Graph of y=p(x) intersects the x-axis at one and only one point. Hence, polynomial p(x) has one and only one real zero.

(iii) Graph of y=p(x) intersects the x-axis at 3 points. Hence, polynomial p(x) has 3 zeros.

(iv) Graph of y=p(x) intersects the x-axis at 2 points. Hence, polynomial p(x) has 2 zeros.

(v) Graph of y=p(x) intersects the x-axis at 4 points. Hence, polynomial p(x) has 4 zeros.

(vi) Graph of y=p(x) intersects the x-axis at 1 point and touches the x-axis at 2 points. Hence, p(x) has 3 zeros.

5.0Benefits of Studying NCERT Solutions Class 10 Maths Chapter 2 Exercise 2.1

• Strengthened foundational concepts 
• Improved problem-solving skills
• Exam preparation and confidence
• Enhanced understanding of key topics
• Easy access and free download