## NCERT Solutions for Maths Chapter 2 Polynomials Class 9 - Free PDF Download

NCERT Solutions for Class 9 Maths Chapter 2 - Polynomials

## FAQs on NCERT Solutions for Class 9 Maths Chapter 2 - Polynomials

**1. What is Polynomial and How is It Classified on the Basis of the Number of Terms and Degrees ? **

A polynomial in a variable x is an algebraic expression of the form

p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + . . . . .+ a₂x²+ a₁x + a₀ ,where a₀, a₁, a₂, . . . . ., aₙ are constants and aₙ ≠ 0.

In the above expression of polynomial p(x), a₀, a₁, a₂, . . . . ., aₙ are respectively the coefficients of variables x⁰, x, x² , . . . . ., xⁿ, and the degree of polynomial p(x) is the non-negative (i.e. n0) power n to which the variable x is raised in the expression.

Each of the expressions aₙxⁿ + aₙ₋₁xⁿ⁻¹ + . . . . .+ a₂x²+ a₁x + a₀ is called terms of the polynomial p(x).

Polynomial class 9 are classified on the basis of the number of terms and degrees they have as follows:

Classification of Polynomials Based on the Number of Terms:

**Monomial:**A polynomial having only one term is known as a monomial.

**Example:** 2x², – 3, –3/2 (y)

**Binomial:**A polynomial having two terms is known as a binomial.

**Example:** 2x² + 1, x – 3, –3/2(y) + x

**Trinomial:**A polynomial having three terms is known as a trinomial.

**Example:** x² - 2x + 2

Classification of Polynomials Based on the Degrees of Its Variable:

**Linear polynomial:**A polynomial having degree one is known as a linear polynomial.

**Example:** 2x + y, x – 3

**Quadratic polynomial:**A polynomial having degree two is known as a quadratic polynomial.

**Example:** 2x² + 1, x² - 2x + 2

**Cubic polynomial:**A polynomial of degree three is known as a cubic polynomial.

**Example:** x³ + 4x² + 7x - 3

**2. Give the Difference between Remainder and Factor Theorem?**

The remainder theorem states that if a polynomial p(x) is divided by (x - a), then the remainder is obtained by evaluating the expression p(a).

While, the Factor theorem states that (x - a) will be a factor of polynomial p(x) only if the remainder obtained by evaluating the expression p(a) equals zero i.e; p(a) = 0.

For Example: let p(x) = x² - 6x + 9, then find the remainder when it is divided by (x - 1).

So, according to the remainder theorem, remainder is obtained by evaluating the expression p(1)

p(1) = 1² - 6(1) + 9

= 1 - 6 + 9 = 4.

Therefore, the remainder is 4 when p(x) = x² - 6x + 9 is divided by (x - 1).

Now, check whether (x - 3) is the factor of p(x) = x² - 6x + 9.

So, according to factor theorem (x - 3) is the factor of p(x) = x² - 6x + 9 only if p(3) = 0.

p(3) = 3² - 6(3) + 9

= 9 - 18 + 9 = 0.

Therefore, (x - 3) is the factor of p(x) = x² - 6x + 9.

**3. What is an Algebraic Identity and What are Its Uses?**

An algebraic identity is an algebraic equation that is true for any values of the variables occurring in it.

Algebraic identities are used to factorize the algebraic expressions and to compute the products of some arithmetic expressions without multiplying them directly.

Most commonly used algebraic identities are, (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b², etc.

**4. What are Polynomials in One Variable?**

Polynomials are algebraic expressions in which variables and constants terms are connected by various arithmetic operators. For ‘Polynomials in one variable’ the terms of the polynomial have the same common variable, with numeric coefficients. In ‘Polynomials in one variable’, the variables are raised to powers and the degree of the equation can be determined with the highest power of the variable. Also, the degree of a polynomial is always a positive integer. Hence, if there is a variable term raised to a negative power in an algebraic expression, then it is not counted as a polynomial.

Examples of Polynomials: 2x³ - 7x² + 3x - 2, in this algebraic expression, there are four terms, and all the variable terms have positive integer exponents. So, this algebraic expression is a polynomial.

**5. What do We Learn in the Class 9 Maths Chapter 2 Polynomials?**

‘Polynomials’ is one of the most important chapters in the Maths syllabus of class 9. The basic highlights of this chapter are listed below.

Polynomial is an algebraic expression consisting of variables, preceded by coefficients, and connected by arithmetic operators. In this chapter, you will learn about polynomials in one variable, that is, the variable coefficient for all the terms in a polynomial, here, will be the same.

The variables along with their coefficients in the algebraic expression are called terms of the polynomial. Some variable terms in a polynomial are raised to exponents.

The degree of a polynomial refers to the greatest exponent of its variable term. If there is a constant term in the polynomial, then it has to be assumed that the exponential value for its variable coefficient is zero.

**6. What are the Important Topics Discussed in Class 9 Maths Chapter 2 Polynomials?**

Remainder theorem, degree of polynomials, factorization, zeros of a polynomial, algebraic identities, etc., are explained in this chapter. These concepts form the basis of higher mathematics so you must have a good knowledge of these concepts. For example, factorization is one of the most basic concepts of algebra, and you will find its application in other chapters as well.

You may expect sums for finding the degree of polynomials to carry fewer marks in the examination, and mostly these sums are in the compulsory part of the question paper. The sums of the NCERT maths book of class 9 are solved in a step-by-step manner on Vedantu and you may refer to it for a better understanding of all concepts of ‘Polynomials’.

7. How do I identify the degree of a polynomial?

The degree of polynomials in one variable is the highest power of the variable in the algebraic expression.

For example, X^2+5X+3. The degree of this equation is 2.

**8. Are NCERT Solutions for Class 9 Maths Chapter 2 difficult to learn?**

No, it is easy to score high in your exams if you regularly practice with Vedantu’s NCERT solutions for Class 9 Maths Chapter 2. The solutions are formulated by experienced subject matter experts who understand the CBSE curriculum. The solutions teach you how to present your answers in exams. Every minute detail is covered in a simple way for students to grasp the concepts easily. The solutions are updated according to the latest guidelines of the CBSE board.

**9. What is a Polynomial?**

Polynomials are expressions that contain one or more terms with coefficients that are not zero. Polynomial terms can be variables, constants, or both. A polynomial's exponents should always be whole numbers. The degree of a polynomial is the highest power in the polynomial. Polynomials are another way to represent real numbers. Further information may be found in Vedantu's NCERT answers for Class 9 Mathematics Chapter 2.

**10. What is the Remainder Theorem?**

When a polynomial p(x) is divided by the linear polynomial (x-a), then the remainder comes out to be p(a). The degree of the polynomial should be greater than one. For more details, refer to Vedantu’s NCERT solutions for Class 9 Maths Chapter 2. The solution PDFs and other study materials such as important questions and revision notes can also be downloaded from the Vedantu app as well for free of cost.

**11. What are the conditions for an expression to be Polynomial?**

The following conditions are important to consider an expression as a polynomial:

All the coefficients should be real numbers.

The exponent should not be a negative number.

For more details, refer to Vedantu’s NCERT solutions for Class 9 Maths Chapter 2. The solution PDFs and other study materials such as important questions and revision notes can also be downloaded from the Vedantu app as well for free of cost.