POLYNOMIALS IN ONE VARIABLE

Published on: March 26, 2024

Introduction

We know that an expression involving variables, constants, and operators (+,−,×,÷) is called an algebraic expression.

Examples: -x2, 5x + y, 2x + 1x + x3
 

Polynomial: An algebraic expression in which the power of variables is a non-negative integer is called a polynomial.

 x3 + x0 is a polynomial, but x-2 + 5  and x + 2x2 + 6 is not a polynomial.

Polynomials in One Variable

Consider a square with sides a. The area of the square is a2. This is an algebraic expression.

Similarly, we are familiar with many algebraic expressions such as 5a, a3 + 2a + 6, 9a2 + 4a, 2a + 5. Such expressions are called polynomials in one variable. Here the variable is ‘a’

Generally, polynomials in one variable are the polynomials of the form axm + b where a and b are constants, x is a variable, and m is a positive integer. A polynomial can be denoted as p(x) or q(x) or f(x) etc. if it contains the variable x

Terms of a polynomial: A term of a polynomial is of the form axm where a is a constant, x is a variable, and m is any positive integer. A polynomial can have only a finite number of terms.

Examples:

1) In the polynomial 2r2 + 25r, there are two terms 2r2  and 25r

2) The polynomial 5x2 + 4x + 2 has three terms: 5x2, 4x and 2
 

Coefficient of a term: It is the constant attached to a term in a polynomial.

Example: In the polynomial 3x2 - 17, the coefficient of x2 is 3 and the coefficient of x0 is 17

 

Constant polynomial: This is a polynomial without any variable, i.e., a polynomial has only one term, which is a constant.

Examples: 2, -35, 26 etc.

 

Zero polynomial: The constant polynomial 0 is known as the zero polynomial.


 

Degree of a polynomial: The highest power of the variable in a polynomial is called the degree of the polynomial.

Examples:

  1. Consider the polynomial f(x) = 2x6 - 3x4 + x + 5 The term with the highest power of x is 2x6. The exponent of x in this term is 6. So, the degree of the polynomial is 6.

  2. Similarly, in the polynomial f(y) = 3y5 - 8y2 - 6, the term with the highest power of y is 3y5. The exponent of y in this term is 5, and thus the degree of the polynomial is 5

Note

  • The degree of a non-zero constant polynomial is zero.

  • The degree of a zero polynomial is not defined.

Example 1

Find the degree of the polynomials 5x5 + 4x + 2 and 3x3 + 2x + 5

Solution:

  1. 5x5 + 4x + 2

The highest power of the variable is 5, and hence the degree of the polynomial is 5

      2. 3x3 + 2x + 5

The highest power of the variable is 3, and hence the degree of the polynomial is 3

Example 2

Consider the given polynomial p(x) and answer the following questions:

p(x) = 3x7 + 3x3 - 3x2 + 19

  1. Find the number of terms in p(x).

  2. Write the coefficients of eq and eq in p(x).

  3. Find the degree of p(x).

Solution:

  1. There are 4 terms in p(x): 3x7, 3x3, -3x2 and 19

  2. The coefficient of x0 19, and the coefficient of x2 is -3

  3. The degree of p(x) is 7 since the highest power of the variable x is 7

 

Classification of polynomials based on the number of terms

A term of a polynomial is of the form axn where a is a constant, x is a variable, and n is any non-negative integer. A polynomial can have a finite number of terms.

The given figure shows the classification of polynomials based on the number of terms.

Examples:

Classify the following polynomials as monomial, binomial, or trinomial

1. x2 + y22. 2y - 3y + 4y33. p2q + pq24. 9pqr

Solution:

1. x2 + y2 : Binomial2. 2y - 3y + 4y3 : Trinomial3. p2q + pq2 : Binomial4. 9pqr : Monomial
 

Classification of polynomials based on the degree of polynomials

The polynomial of degree n is an expression of the form

an xn + an-1 xn-1 + ....... + a1x + a0 where a0, a1, a2 ..... an are constants and an 0

If all the constants a0, a1, a2 ........ an = 0, a zero polynomial is obtained, which is denoted by 0. The degree of a zero polynomial is not defined.

The given figure shows the classification of polynomials based on the degree of polynomials:

Examples:

(a) p(x) = x2 + 2x - 19(b) q(x) = x3 - 3x(c) r(x) = x + 3(d) s(x) = x3 - 5x - 22

Classify the following polynomials as linear, quadratic, and cubic polynomials

Solution:

Linear Polynomial : r(x) = x + 3

Quadratic Polynomial: p(x) = x2 + 2x - 19

Cubic Polynomial: q(x) = x3 - 3xs(x) = x3 - 5x - 22