Polynomials

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:- 2x2 + x + 5

In above examples, 2x2, 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) 2x2 + x + 5

Coefficient of x2 = 2 (2 is multiply to x2)

Coefficient of term x = 1(2 is multiply to x2)

5 is a coefficient of x0 (x0 =1) and known as constant polynomial.

(ii) -2x2 – x – 5

Coefficient of x2 = -2 (-2 is multiply to x2)

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

-5 is a coefficient of x0 (x0 =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, 4x2 or 2

(ii)Binomial-

Polynomials having two terms only are known as binomial.

Eg:-  4x2  + 2x (4x2  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:-  4x2  + 2x  -5(4x2, 2x and – 5  are three terms)

x2 –  2x – 7(x2, -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) 4x7 + 2x2 – 5x + 6

The degree of the polynomial is 7.

(ii) 4x11 + 2x10 – 5x9 + 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 x0.

The value of x0 =1

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

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

4x7 + 2x2 – 5x + 6

Ans- The constant polynomial is 6.

6 is a coefficient of x0 [ x0 =1]

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

6 = 6 x0

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) = x3 + 1 by x +1

Ans-

Finding remainder by division method

 

Helping Topics

NCERT Solutions Class 9

Worksheet Class 9

Algebraic expressions

 

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