# Degree of an Algebraic Expression

The Algebraic Expressions are made up of Constants and Variables. In an Algebraic Expression, there may be one variable or multiple variables. The general definition for the Degree of an Algebraic Expression is the highest Exponent or Power of variable or variables present in the expression. However, the methods for finding out the degree of an Algebraic Equation having exactly one variable or multiple variables are different as discussed below.

## Degree of an Algebraic expression with one variable

The Degree of an Algebraic Expression having exactly one variable may be defined as the highest Exponent or Power of the variable that is present in the Algebraic Expression.

### Examples of Degree of an Algebraic Expression with one variable

To understand the concept of the Degree of an Algebraic Expression, let us consider the following examples,

• First, let us consider a simple Algebraic Expression as $(x + 2)$.

In the above example, the Algebraic Expression $(x + 2)$ has two terms i.e., $x$ and $2$. Only, the first term has one variable which is $x$ and the Exponent or Power of the variable is $1$. Therefore, the Degree of the given Algebraic Expression is $1$.

• Let us consider an another Algebraic Expression as $(x^2+ 4x + 3)$.

The above Algebraic Expression also has only one variable i.e., $x$. However, the variable is present in both the first and the second term. In such cases, to find out the Degree we have to consider the variable having the Highest Power. In the first term $x^2$, the power of $x$ is $2$ and in the second term $4x$, the power of $x$ is $1$. Therefore, the Degree of the given Algebraic Expression is $2$.

• $x^5 + 3x^2 + 9$

Similarly, in the above Algebraic Expression, the first term $x^5$ has the Power of $5$, and the second term $3x^2$ has the power of $2$. Therefore, the Degree of the above Algebraic Expression is the highest Power of $x$ which is $5$.

## Degree of an Algebraic Expression having more than one variables

As discussed in the above section, an Algebraic Expression may have multiple variables present in a single term or distributed in different terms. The Degree of an Algebraic Expression having more than one variable or multiple variables may be defined as the Highest Sum of Exponents or Powers of different variables present in any term of the Algebraic Expression.

### Examples for the Degree of an Algebraic Expression having more than one variables

To understand the concept of the Degree of an Algebraic Expression having multiple variables, let us consider the following examples.

• Let us consider an Algebraic Expression as, $x^4 + 3xy^2 + 2$

In the given Algebraic Expression, the first term $x^4$ has only one variable that is $x$ and the power of $x$ is $4$.

However, the second term, $3xy^2$ has two variables i.e., $x$ and $y$. The power of $x$ in the second term is $1$ and the power of $y$ is $2$. Now, the sum of the powers of these two variables is $(1+2)=3$. Therefore, the power of the second term becomes $3$.

The Highest Power in the above example i.e., $4$ is hold by the first term $x^4$. Therefore, the Degree of the given Algebraic Expression is $4$.

• $x^2 + 3x^2y^3 + 2$

In the above example of Algebraic Expression, the power of the first term $x^2$ is $2$.

The power of $x$ and the power of $y$ in the second term $3x^2y^3$ are $2$ and $3$ respectively. The sum of the powers of $x$ and $y$ in the second term is $(2+3)=5$.

The Highest Power in the above example i.e., $5$ is hold by the second term $3x^2y^3$. Therefore, the Degree of the given Algebraic Expression is $5$.

## Solved Examples

### Q1: Find the degree of the Algebraic Expression $x^2+4xy+2$?

Solution: To find the degree of the given expression let us first find the power of each term separately.
Power of the first term = Power of $x$ = $2$
Power of the Second term = Power of $x$ + Power of $y = 1 + 1 = 2$
The third term is a constant.
From above, the power of the first and the second term is similar which is $2$. Therefore, the degree of the expression is $2$.

### Q2: Find the degree of the Expression $x^5+3x^2y^4+8z$?

Solution:
Power of the First term = power of $x$ = $5$
Power of the Second term = power of $x$ + power of $y$ = $2+4$=$6$
Power of the Third term = Power of $z$ = $1$
The second term has the highest sum of powers that is $6$. Therefore, the degree of the expression is $6$.

### Q3: Find the degree of the Expression $x^3y^4+2x^3y^2z^2+8zy^2$?

Solution:
Power of the First term = power of $x$ + power of $y$ = $3+4=7$
Power of the Second term = power of $x$ + power of $y$ + power of $z$ = $3+2+2$=$7$
Power of the Third term = Power of $z$ + Power of $y$ = $1+2=3$
The first and the second term has the highest sum of powers that is $7$. Therefore, the degree of the expression is $7$.