NCERT Solutions for Class 9 Maths Chapter 2 - Polynomials

• Exercise 2.1: This exercise covers the definition and basic concepts of polynomials. The questions in this exercise aim to familiarise students with terms like coefficients, exponents, degrees, and standard forms of polynomials. Students are also required to classify polynomials based on their degrees. They will have to solve problems related to the addition, subtraction, and multiplication of polynomials and also learn how to factorise polynomials.

• Exercise 2.2: This exercise deals with the factors and zeros of polynomials. The questions in this exercise require students to find the factors and zeros of given polynomials. They will also learn how to use the factor theorem and remainder theorem to factorise polynomials and find their zeros.

• Exercise 2.3: This exercise covers the division algorithm for polynomials. The questions in this exercise require students to divide a polynomial by another polynomial using the long division method. They will also learn how to use the remainder theorem to find the remainder when a polynomial is divided by another polynomial.

NCERT Solutions for Class 9 Maths Chapter 2 Polynomial - PDF Download

Variables - The Unknown Value

Have you ever wondered why children have different heights? Some children grow taller and some end up being shorter than average. To answer this question Scientists have come closer and researched the parameters in the form of variables that are the cause of height.

The word ‘variable’ is derived from the word ‘vary’ which means changing. Therefore, a variable can be any trait, condition, or factor that can change by only differing amounts or it is the unknown term whose value is not known. Example: A child’s height is dependent on the amount of protein and nutrients he or she consumes. Not only that, the height of kids is also dependent on their DNA which means if their parents are tall then there are more chances of them being tall whereas short parents usually have short kids. The height of the kid is also dependent on the rate of work or activities. It is believed that children with more activities like jumping, running, skipping, etc tend to grow faster. Thus, nutrients, DNA, and activities are the three variables that control the height in our body. These variables keep changing from body to body.

For example, while cooking dal we know that the quantity of water is thrice the number of lentils. That you can add 1 cup of lentils to three cups of water. This process can be expressed as,

“3x + x”

Here, the quantity of lentils is variable. That means if the quantity of lentils changes then the quantity of water also changes.

In a World Full of Variables, You Will Always Find Constant.

There is one interesting thing about constants and that is this it never changes. A constant is actually a value that is a fixed number on its own. For example - In the equation 9 - x = 5, 9 and 5 are two constants whose values will not change whereas the value of x is not known. Thus, x is a variable.

Can Constant be a Coefficient To?

Since now we already know about variables, it is easier for us to understand the constant and coefficient. A coefficient is usually the number that is multiplied by the variable or letters. For example in ‘5x + y - 7’, 5 is a coefficient of x in the term 5x because it is a number that is multiplied by the unknown variable x. Also, in the term y, it can be considered as the coefficient of y because y can be written as 1xy.

The coefficient is the number that is always multiplied by the variables but constants are terms without variables. Therefore, coefficients cannot be called constants and vice versa. In the aforementioned example, -7 is constant.

Like Terms

Like terms are the terms having the same variables raised to the same power. In 5x + y - 7, no variable is common therefore no like terms. 

In 5a + 2b - 3a + 4 the terms like 5a and -3a are like terms whereas 4 is constant.

What is a Polynomial?

The word Polynomial is derived from the word poly ("many") and nominal ("term"). It is an expression consisting of many terms such that each term holds at least one variable. The variables can be raised to the power and further multiplied by a coefficient but the simplest polynomials hold one variable. The terms are separated by signs( + or - ). Also, the variables and numbers can be combined using addition, subtraction, multiplication or division but it can never be divided by a variable which means a term can never be like  2/x. A polynomial can also not have infinite terms. It always has a finite sum of terms with all variables having whole-number exponents and no variable as a denominator.

Polynomials are composed of the following:

• Constants such as 3, −20, or ½, etc.

• Variables such as g, h, x, y, etc.

• Exponents such as 2 in y2 or 5 in x5  etc

Examples of Polynomials: 5x3 – 2x2 + x – 13 and  x2y3 + xy.

Degree of a Polynomial

It is simply the highest of the powers or exponents on the terms present in the algebraic expression.

Example: In 7x – 5, the first term is 7x, whereas the second term is -5. The power on the variable of the given first term is one and on the second term is zero. Since the highest exponent is one, the degree of the polynomial is also 1.

Types of Polynomials

Polynomials can be classified on the basis of

1. Number of Terms.

2. Degree of a polynomial.

Classification on the Basis of Terms

A polynomial either has one term, two terms, three terms, or more than three terms.

1. Monomials- ‘Mono’ stands for one and ‘mial’ stands for terms thus an algebraic expression with one term is called a monomial. 

2. Binomials- ‘Bi’ stands for two and ‘mial’ stands for terms therefore an algebraic expression with two, unlike terms is called binomials. 

3. Trinomials- ‘Tri’ stands for three and ‘mial’ stands for terms thus an algebraic expression with three unlike terms is called trinomials.

Classification on the Basis of Degrees

The Degree of Polynomial is considered as the highest value of the exponent in the expression because it is the largest exponent. We can also call it an order of the polynomial. While finding the degree of the polynomial, remember that the polynomial powers of the variables must be either in an ascending or descending order.

1. Linear Polynomial: If the expression holds degree 1 then we can call it a linear polynomial. 

2. Quadratic Polynomial: If the expression holds degree 2 then it can be called a quadratic polynomial.

3. Cubic Polynomial: If the expression holds degree 3 then it will be called a cubic polynomial.

Zeros of Polynomials

If the value of every coefficient of a variable is zero then it is called the zeros of a Polynomial. In order to find the relationship between the zeroes and coefficients of a given quadratic polynomial, we can find the zeros of the polynomial by the factorization method that is, by taking the sum and product of these zeros.

Operations on Polynomial

There are four main polynomial operations which are:

• Addition of Polynomials

• Subtraction of Polynomials

• Multiplication of Polynomials

• Division of Polynomials

NCERT Solutions Class 9 Maths Chapter 2 Polynomials All Exercises

Conclusion

NCERT Maths Class 9 Solutions Vedantu's polynomials provide a thorough grasp of this significant subject. Students can build a solid foundation in algebra by concentrating on important ideas such as polynomial expressions, degree of polynomials, and polynomial operations.It's important to pay attention to the step-by-step solutions provided in the NCERT Solutions, as they help clarify concepts and reinforce problem-solving techniques. Understanding polynomials is crucial as they form the basis for understanding more complex algebraic concepts. Approximately four to five questions from this chapter have usually been included in previous year's question papers. As a result, practicing a range of issues from NCERT Solutions and past test papers helps improve exam readiness and confidence.

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Chapter-Specific NCERT Solutions for Class 9 Maths

Given below are the chapter-wise NCERT Solutions for Class 9 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.