Polynomials in One Variable

In Mathematics, a polynomial is an algebraic expression with one or more than one term with a non-zero coefficient. The polynomial expression is made up of variables, coefficients, and constants which are connected by a mathematical operator. In this article, we will learn the definition of polynomials in one variable, terminologies related to polynomials, and the classification of polynomials in one variable based on its degree with many solved examples.

Table of Contents:

• Polynomials in One Variable Definition
• Polynomials – Related Terminologies
• Classification of Polynomials Based on Degree
• Solved Examples
• FAQs

Polynomials in One Variable Definition

In Mathematics, a polynomial is an expression that consists of variables, coefficients and constants, which are connected by mathematical operations, such as addition, subtraction, multiplication and division. “Polynomials in one variable is an algebraic expression that consists of one variable in it.” Some of the examples of polynomials in one variable are given below:

• x+2
• x²+3x+2
• m³+2m²-m

Polynomials – Related Terminologies

The different terms related to polynomials are given below:

Terms: In an expression, a term can be a variable or constant or a product of variable and constant.

Coefficient: A coefficient is a numerical value, which is written along with a variable.

Variable: A variable is a letter that represents the unknown value in an expression.

Constant: A constant is a number, whose value never changes in an expression.

Consider an example, 5x+2

Here,

The variable is x, coefficient is 5, constant is 2, and terms are 5x and 2.

Classification of Polynomials in One Variable Based on Degree

The polynomials in one variable can be classified based on the degree of a polynomial. Before discussing the classification, let us have a look at the degree of a polynomial.

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

For example, 5x²+ 2x+7

The degree of a polynomial is 2, as the highest power of the variable “x” in a polynomial is 2.

(Note: If a polynomial has more than one variable, the degree of a polynomial is the highest sum of different variables in any of the terms)

Based on the degree of a polynomial, the polynomials in one variable are classified as follows:

• Zero polynomial or constant polynomial
• Linear Polynomial
• Quadratic Polynomial
• Cubic Polynomial

Zero Polynomial or Constant Polynomial

If the degree of the polynomial is zero (0), then the polynomial is called zero or constant polynomial. Such kinds of polynomials have only constants. They don’t have variables.

The examples of constant polynomials are 2, 5, 7 and so on.

Here, 2 can be written as 2x⁰, 5 can be written as 5x⁰, and so on.

Linear Polynomial

If the degree of the polynomial is 1 (one), then the polynomial is called a linear polynomial. The linear polynomial in one variable has only one solution.

Examples of linear polynomials in one variable are:

• m+2
• y+5
• x+10

Quadratic Polynomial

A polynomial with the highest degree of 2 is called a quadratic polynomial. A quadratic polynomial in one variable has only two solutions. Some of the examples of quadratic polynomials in one variable are:

• 9x² – 10
• x² +5x+9
• m²+25

Cubic Polynomial

If the highest exponent of a variable in a polynomial is 3 (i.e. degree of a polynomial is 3), then the polynomial is called a cubic polynomial. A cubic polynomial in one variable has exactly 3 solutions. The examples of a cubic polynomial in one variable are:

• 7x³– 21
• 8x³+2x+9
• 10m³ + (5/4)

Solved Examples

Example 1:

Find the degree of the polynomial in one variable.

1. x⁸+x⁵+3x+10
2. 9
3. y⁵+y²-7

Solution:

(1) x⁸+x⁵+3x+10: The degree of a polynomial is 8, as the highest degree in the given polynomial is 8.

(2) 9: The degree of the polynomial is 0. The given polynomial is called constant polynomial, as 9 can be represented as 9x⁰.

(3) y⁵+y²-7: The degree of the polynomial is 5. The highest exponent of the variable in a polynomial is 5.

Example 2:

Find the coefficient of x in the polynomial 3x²+5x+10.

Solution:

The coefficient x in the polynomial 3x²+5x+10 is 5.