**Notes of chapter: Polynomials are presented below. Indepth notes along with worksheets and NCERT Solutions for Class 9.**

**(1) ****Polynomial-**

**Polynomial **is an algebraic sum that consists of terms [variables, constants and operations], variables can be in positive integer exponentiation (raise to the power) only and variables can not appear in denominators and under radical sign.

**(2) Terms of the polynomials-**

**Terms of the polynomials** are the parts of the polynomials that are separated by addition or subtraction.

**Eg:-** 2x^{2} + x + 5

In above examples, 2x^{2}, x and 5 are terms that make a polynomial.

**(3)Coefficient- **

The number that is multiply to a variable is called **coefficient** of each term.

**Eg:- **

**(i)** 2x^{2} + x + 5

Coefficient of x^{2} = 2 (2 is multiply to x^{2})

Coefficient of term x = 1(2 is multiply to x^{2})

5 is a coefficient of x^{0} (x^{0} =1) and known as constant polynomial.

**(ii)** -2x^{2} – x – 5

Coefficient of x^{2} = -2 (-2 is multiply to x^{2})

Coefficient of term x = -1(-1 is multiply to x)

-5 is a coefficient of x^{0} (x^{0} =1) and known as constant polynomial.

**(iii)Zero polynomial –**

The constant polynomial 0 is called the **zero polynomial**.

**(4)** **Types of polynomial** **on the basis of terms**

**(i)Monomial-**

Polynomials having one term only are known as **monomial**.

**Eg:-** 2x, 4x^{2} or 2

**(ii)Binomial-**

Polynomials having two terms only are known as **binomial.**

**Eg:-** 4x^{2} + 2x (4x^{2} and 2x are two terms)

x + 2(x and 2 are two terms)

**(iii) Trinomial-**

Polynomials having three terms only are known as **trinomial.**

**Eg:-** 4x^{2} + 2x -5(4x^{2}, 2x and – 5 are three terms)

x^{2} – 2x – 7(x^{2}, -2x and -7 are three terms)

**(5)Degree of the polynomial-**

The highest power of the variable of the polynomial is known as **degree of the polynomial.**

**Eg:-**

**(i)** 4x^{7} + 2x^{2} – 5x + 6

The degree of the polynomial is 7.

**(ii)** 4x^{11} + 2x^{10} – 5x^{9} + 7x – 2

The degree of the polynomial is 11.

**(iii)The degree of** **a non – zero constant polynomial is zero.**

The constant of a polynomial is a coefficient of x^{0}.

The value of x^{0} =1

Therefore, the degree of a non zero constant polynomial is zero.

**Eg:-** Find the degree of constant polynomial in given polynomial.

4x^{7} + 2x^{2} – 5x + 6

**Ans-** The constant polynomial is 6.

6 is a coefficient of x^{0} [ x^{0} =1]

Therefore, we can write 6 in multiplication of variable with powers.

6 = 6 x^{0}

Hence, the degree of 6 is zero.

**(9)Remainder theorem-**

**Remainder theorem** is used to avoid long division in polynomials to get remainder.

Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x-a, then the remainder is p(a).

**Eg:-** find the remainder on dividing p(x) = x^{3} + 1 by x +1

**Ans-**

Finding remainder by **division method**

**Helping Topics**