NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials

NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials pdf








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All Ncert Math Class 9 solutions are available for free download here. Math NCERTs are generally required in all CBSE schools, although even some state board schools follow them. The Ncert solution for 9th grade math is important because it lays the foundation for building upper grade math. Unlike eighth grade math, the ninth grade math curriculum is very extensive and feels a bit difficult.

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Polynomials


Polynomials are algebraic expressions that contain indeterminates and constants. You can think of polynomials as a dialect of mathematics. They are used to express numbers in almost every field of mathematics and are considered very important in certain branches of math, such as calculus. For example, 2x + 9 and x2 + 3x + 11 are polynomials. You might have noticed that none of these examples contain the "=" sign. Have a look at this article in order to understand polynomials in a better way.


What is a Polynomial?


A polynomial is a type of expression. An expression is a mathematical statement without an equal-to sign (=). Let us understand the meaning and examples of polynomials as explained below.

Polynomial Definition


A polynomial is a type of algebraic expression in which the exponents of all variables should be a whole number. The exponents of the variables in any polynomial have to be a non-negative integer. A polynomial comprises constants and variables, but we cannot perform division operations by a variable in polynomials.

Polynomial Examples


Let us understand this by taking an example: 3x2 + 5. In the given polynomial, there are certain terms that we need to understand. Here, x is known as the variable. 3 which is multiplied to x2 has a special name. We denote it by the term "coefficient". 5 is known as the constant. The power of the variable x is 2.

Standard Form of Polynomials


The standard form of a polynomial refers to writing a polynomial in the descending power of the variable.

Example: Express the polynomial 5 + 2x + x2 in the standard form.

To express the above polynomial in standard form, we will first check the degree of the polynomial.

In the given polynomial, the degree is 2. Write the term containing the degree of the polynomial.
Now, we will check if there is a term with the exponent of variable less than 2, i.e., 1, and note it down next.
Finally, write the term with the exponent of the variable as 0, which is the constant term.
Therefore, 5 + 2x + x2 in standard form can be written as x2 + 2x + 5.

Always remember that in the standard form of a polynomial, the terms are written in decreasing order of the power of the variable, here, x.

Terms of a Polynomial


The terms of polynomials are defined as the parts of the expression that are separated by the operators "+" or "-". For example, the polynomial expression 2x3 - 4x2 + 7x - 4 consists of four terms.

Like Terms and Unlike Terms
Like terms in polynomials are those terms which have the same variable and same power. Terms that have different variables and/or different powers are known as unlike terms. Hence, if a polynomial has two variables, then all the same powers of any ONE variable will be known as like terms. Let us understand these two with the help of examples given below.

For example, 2x and 3x are like terms. Whereas, 3y4 and 2x3 are unlike terms.


Degree of a Polynomial


The highest or greatest exponent of the variable in a polynomial is known as the degree of a polynomial. The degree is used to determine the maximum number of solutions of a polynomial equation (using Descartes' Rule of Signs).

Example 1: A polynomial 3x4 + 7 has a degree equal to four.

The degree of the polynomial with more than one variable is equal to the sum of the exponents of the variables in it.

Example 2: Find the degree of the polynomial 3xy.

In the above polynomial, the power of each variable x and y is 1. To calculate the degree in a polynomial with more than one variable, add the powers of all the variables in a term. So, we will get the degree of the given polynomial (3xy) as 2.

Similarly, we can find the degree of the polynomial 2x2y4 + 7x2y by finding the degree of each term. The highest degree would be the degree of the polynomial. For the given example, the degree of the polynomial is 6.




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