The highest non-negative integer exponent of a variable in a single variable polynomial is called the degree of a polynomial in one variable.
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.
Let us learn how to find the degree of a polynomial in one variable from the following understandable examples.
$(1).\,\,\,$ $3x^2$
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 $1$.
$(1).\,\,\,$ $4x^{-3}+7x-6$
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.
$(1).\,\,\,$ $6$
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.
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