Degree of a Polynomial in a single variable

The highest non-negative integer exponent of a variable in a single variable polynomial is called the degree of a polynomial in one variable.

Introduction

A polynomial in one variable is formed in mathematics by either a single variable expression or connecting many single variable expressions. Every variable in an expression consists of an exponent and it is a factor to determine the degree of a polynomial in one variable.

The following two factors are the rules for finding the degree of any polynomial in a single variable.

1. If a polynomial in one variable is formed by a single expression, then the exponent of the variable is considered as the degree of the polynomial in one variable. If it is formed by connecting many expressions, then the highest power (or exponent) of the variable is considered as its degree.
2. The exponent of the variable should be a non-negative integer.

Examples

Let us learn how to find the degree of a polynomial in one variable from the following understandable examples.

Polynomials

$(1).\,\,\,$ $3x^2$

It is an expression, which consists of only one term. In this expression, $x$ is a variable and its exponent is $2$. The exponent $2$ is a non-negative integer. Hence, it is called a polynomial and its degree is $2$.

$(2).\,\,\,$ $-x^3+7x^2-5x+9$

Four expressions in one variable are connected to form this expression. The exponents of the variables in the terms are $3$, $2$, $1$ and $0$ respectively. All of them are non-negative integers. Therefore, the above expression is a polynomial in one variable. The highest exponent of variable in this expression is $3$. Hence, the degree of this polynomial is $3$.

$(3).\,\,\,$ $7y^4-6y$

Two expressions in a variable $y$ are connected to form this expression. In this expression, the exponents of the variable $y$ are $4$ and $1$ respectively. The two indices are non-negative integers. Hence, the expression is a polynomial in one variable. The highest exponent of the variable is $4$. So, the degree of this polynomial is $4$.

$(4).\,\,\,$ $4z+\sqrt{3}$

This expression is formed by connecting to two expressions. In this expression, the exponents of variable $z$ are $1$ and $0$, which are non-negative integers. Therefore, the expression is a polynomial in one variable $z$ and its degree is $1$.

Expressions

$(1).\,\,\,$ $4x^{-3}+7x-6$

It is an expression, which consists of three terms. In this expression, the exponents of the variable are $-3$, $1$ and $0$. Due to the negative exponent, the expression cannot be called a polynomial in one variable.

$(2).\,\,\,$ $4\sqrt{y}$

This expression can be written as follows.

$\,\,\,=\,\,\,$ $4y^{\Large \frac{1}{2}}$

The exponent of the variable $y$ is not an integer. So, the expression is not a polynomial.

Constants

$(1).\,\,\,$ $6$

A real number is considered as a polynomial for one reason because it can be written as a polynomial in one variable.

$\implies$ $6 \,=\, 6 \times 1$

$\implies$ $6 \,=\, 6 \times x^0$

$\,\,\,\therefore\,\,\,\,\,\,$ $6 \,=\, 6x^0$

Hence, the degree of this polynomial is zero.

$(2).\,\,\,$ $0$

The number zero belongs to real number group. Hence, it can be a zero-degree polynomial but it is not because the number $0$ can be written as follows.

$\implies$ $0 \,=\, 0 \times x^0 \,=\, 0x^0$

$\implies$ $0 \,=\, 0 \times x^1 \,=\, 0x^1$

$\implies$ $0 \,=\, 0 \times x^2 \,=\, 0x^2$

In the case of zero, the number $0$ is considered as a polynomial in one variable but its degree is undefined.