Notes of chapter: Polynomials are presented below. Indepth notes along with worksheets and NCERT Solutions for Class 9.
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.
The number that is multiply to a variable is called coefficient of each term.
(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
Polynomials having one term only are known as monomial.
Eg:- 2x, 4x2 or 2
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)
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.
(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.
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
Finding remainder by division method