Chapter 2

Polynomials

You have already studied polynomials in class 10 CBSE. Polynomial is an essential subject in the curriculum. The topics include production, geometrical meaning of the zeros of polynomials etc., in Class 10th maths chapter 2 NCERT solutions.

Polynomials are very important topics in mathematics that have application in maths and other subjects such as science. In this article, we will cover most of the concepts of the NCERT solution for class 10th maths polynomials, such as the geometrical meaning of the zeros of polynomials, the relationship between zero, and the coefficient of a Polynomial, etc.

Class 10 maths polynomials is an important chapter that requires thorough preparation and practice. We have fitted our class 10 maths NCERT solutions polynomials to meet the students' needs. In this chapter, many elements of polynomials are covered. The Chapter 2 Class 10th maths NCERT solution format follows a step-by-step procedure that adds comprehended topics.

Topics Covered in this chapter: (content table)

S.no.   Topics
            1.    Introduction to Polynomials
            2.    Geometrical meaning of the zeros of polynomials
            3.    Relationship between zero and coefficient of a                               Polynomials
            4.    Division Algorithms for polynomials
            5.    FAQs

Introduction to Polynomials

A polynomial refers to an algebraic expression that contains two or more terms.  

What do you mean by polynomial? You have studied the polynomial in Class 9th. The Polynomial word is derived from the Greek word "Poly" and "nominal", which means "many" and " terms". So the meaning of a polynomial is "many terms", for example, P(x)=x2+6x-12. A polynomial can be any number of terms but infinite.

Examples of the constants, variable, and exponent are as follows

Constant: 1,4,6,3 etc

Variables: a, b, c, etc

Exponent: a2, b3, d2, etc

Polynomial is defined as an expression that is composed of constants, exponents, and variables. These expressions are involved in the operation of addition, subtraction, and multiplication but not division.

Different types of polynomial based on the terms are as follows:

1. Monomial- 3a2

2. Binomial- 3x2 - 4x

3. Trinomial- 3x2 - 4x + 5

Types of polynomial according to a degree are as follows:

1. Linear Polynomial - where the degree of a monomial is 1. For example -  4a+1

2. Quadratic Polynomial - where the degree of a monomial is 2. For example - 4a2+1a+1

3. Cubic Polynomial - where the degree of a monomial is 3. For example - 6a3+4a+3a+1

4. Quartic Polynomial- where the degree of a monomial is 4. For example - 6a4+3a+3a+2a+1

The real number k is said to be a zero of a polynomial p(x) if p(k)= 0. Polynomials can easily be represented graphically.

The zero of the polynomial is the x-coordinate of the point, where the graph intersects the x-axis. If a polynomial p(x) intersects the x-axis at ( 3, 0), then 3 is the zero of the polynomial.

The graph of a linear polynomial intersects the x-axis at a maximum of one point. Therefore, a linear polynomial has a maximum of one zero.

Quadratic polynomial ax2 + bx +c, has nil, one or two zeroes.

Nil zeros - in the e.g., p(x) = x2 + 1 has no zeros, thus the polynomial does not intersect the x-axis.

One zero -  in the e.g., p(x) = x2 -4x+4 has one zero, thus polynomials intersect at x-axis at 2.

Two zero -  in the e.g., p(x) = x2-4  has 2 zeros that are -2, and 2, thus polynomial does intersect the x-axis at two places, x=2 & x = -2

It has one, two, and three zeros. There can't be any nil zero in the cubic polynomial.

p(x) = x3, in the e.g., There is one zero of the cubic polynomial that is 0 and the graph of this polynomial intersect the x-axis at 0

p(x) = x3 - x2, in the e.g., there are 2 zero of the cubic polynomial that is 0 and 1, thus the graph of this polynomial intersect the x-axis at two places X=0 and X =1

p(x) = x3 - 4x, In the e.g., there are 3 zero of cubic polynomial, that is zero +2 and -2, thus the graph of this polynomial intersect x-axis at three places x=-2,, x= 0, and x= 2

 

Now we will learn the relationship between zeros and coefficients of a Polynomial.

Let’s take a quadratic equation

ax^2+bx+c

In it, there are two zeros named α and β.

In the equation, coefficients are a = coefficient of x^2, b = coefficient of x, c = Constant.

We have to see the relationship between alpha, beta, and a b c.

The linear polynomial can be written as (ax + b)

Zero of a Linear Polynomial

(ax+b) = 0

X= -b/a

Here x is zero, –b/a is coefficient, and x = -b/a is the relationship between zero and coefficient.

A quadratic polynomial is more important than linear from the exam point of view.

Equation of Quadratic Polynomial

ax^2 + bx + c

Sum of zeros :  α + β  = -b/a =  – Coefficient of x/ Coefficient of x2

Product of zeros : αβ = c/a = constant term/coefficient of x^2

In cubic polynomials, there are 3 zeroes.

ax^3 + bx^2 + cx + d

Sum of zeroes in cubic polynomial =

-b/a = -Coefficient of x^2/Cofficient of x^3

Product of zeroes = -d/a = -Constant term/Coefficient of x^2

 

Division of polynomials and Division of numbers are similar. If we divide numbers, we get remainder and quotient, and the remainder is less than the divisor or zero. The relationship of dividend, quotient, and remainder is always satisfied, which is

Dividend = Quotient x Divisor + Remainder

The equation is known as Euclid’s division lemma. Euclid’s division lemma is a function in which the dividend is equal to divisor times the quotient and addition of the remainder.

We divide into the largest place value first, when we divide numbers. Analogously, we divide the largest degree first for polynomials.

 

Step 1 - Take two polynomials. One is the dividend, and the other is the divisor.

Step 2 - Look at the highest power of the variable in the dividend

Step 3 - Now look at the highest power of the variable in the divisor

Step 4 - (Dividend highest power must be greater than divisor highest power of variable) Arrange both dividend and divisor in standard form (highest power variable to lowest power).

Step 5 - Make a bracket and put a dividend in the bracket.

Step 6 - Put the divisor outside.

Step 7 - divide the first term of dividend from the first term of the divisor.

Step 8 – Write the remaining term on another side of the bracket. Now multiply the remaining term to divisor and write the result of multiply below dividend. Change the signs of the result and cancel it with the dividend.

Step 9 – Do the same steps with the remaining dividend.

 

Many practice questions are available for students to solve. Practice questions will prepare you for the exam and help you to take command of the concept.

The questions are from class 10, chapter 2 polynomials, and according to the CBSE syllabus.

Read the questions carefully and answer the questions accordingly.

 

1. What are polynomials class 10th?

Dictionary meaning of polynomial is poly + nomial. “Poly” means many, and “nomial” means terms. The polynomial is included in algebra. In algebra, a term has two parts that are multiplied together. The first part is real numbers, and the second part is variable. You can use variables with power but have to follow some conditions. First is variable power must be a non-negative integer, and variable power must be an integer. Examples of some terms in algebra are 6x^2, 5x^2y. Connection of many terms connected with + or – sign.

2. How many exercises are in class 10 polynomials?

In the class 10 polynomials chapter 2, there are four exercises – 2.1, 2.2, 2.3, 2.4.

3. List out the concepts covered in class 10 polynomials?

Concepts covered in class 10 Maths Chapter 2 polynomials are following:-

· Introduction to Polynomials

· Geometrical Meaning of the Zeros of Polynomial

· Relationship between Zeros and Coefficients of a Polynomial

· Division Algorithm for Polynomials

 

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