On this post we explain what the degree of a polynomial is and how to find the degree of a polynomial. You will also see several examples of how to do it and the different types of polynomials according to their degree. Finally, you will find solved problems to practice.

Table of Contents

## What is the degree of a polynomial?

The definition of degree of a polynomial is as follows:

**The degree of a polynomial is the highest of the degrees of its terms.** Therefore, if the polynomial has only one variable, the degree of the polynomial is the largest exponent to which the variable of the polynomial is raised. On the other hand, if the polynomial has two or more variables, the degree of the polynomial is the largest sum of exponents of its terms.

For example, the degree of the following polynomial is 5 because the maximum value of the exponents of its terms is 5:

## Examples of degrees of polynomials

Once we know how to identify the degree of a polynomial, let’s see more examples to finish understanding its meaning:

- Example of a zero degree polynomial:

- Example of a first degree polynomial:

- Example of a second degree polynomial:

- Example of a third degree polynomial:

- Example of a fourth degree polynomial:

## How to find the degree of a polynomial with two or more variables

We have just seen how to determine the degree of a univariate polynomial (polynomial with a single variable). However, what is the degree of a multivariate polynomial?

First of all, you must remember that if a term has two or more variables, its degree is the sum of the exponents of its variables. For example, the degree of the term 5x^{4}y^{3} is equal to 7, since 4+3=7.

So, **to find the degree of a polynomial with two or more variables,** we first have to calculate the degree of each of its terms, thus, the degree of the polynomial will be the highest degree of its terms.

As an example, we are going to find the degree of the following polynomial with three variables:

The degree of the first monomial of the polynomial is 9 (5+4=9), the second term of the polynomial is of degree 6 (3+2+1=6) and, finally, the third element of the polynomial is of degree 8 (6+2=8). Therefore, the degree of the polynomial of the problem is 9, since it is the maximum degree of its monomials.

## Names of polynomials by degree

According to the degree of the polynomials, we can classify them as follows:

Degree of the polynomial | Name | Example |

Polynomial of degree 0 | Constant (or zero polynomial) | 5 |

Polynomial of degree 1 | Linear polynomial | x+2 |

Polynomial of degree 2 | Quadratic polynomial | x^{2}+3x+1 |

Polynomial of degree 3 | Cubic polynomial | x^{3}-4x+2 |

Polynomial of degree 4 | Quartic polynomial | 5x^{4}+x^{2}+9 |

Polynomial of degree 5 | Quintic polynomial | x^{5}-2x^{3}+3x |

Polynomial of degree 6 | Sextic polynomial | 2x^{6}+8x^{4}-x |

Polynomial of degree 7 | Septic polynomial | -3x^{7}+4x^{4}+3x |

Polynomial of degree 8 | Octic polynomial | x^{8}-9x^{2}+6 |

Polynomial of degree 9 | Nonic polynomial | x^{9}+7x^{5}-x^{2} |

Polynomial of degree 10 | Decic polynomial | 8x^{10}+3x^{8}-4x^{5}-6x^{2} |

## Practice problems on finding the degree of a polynomial

### Problem 1

Find the degree of the following polynomial:

**See solution**

The polynomial has only one variable, therefore the degree of the polynomial is its highest exponent, which is 4.

### Problem 2

What is the degree of the following polynomial?

**See solution**

To find the degree of the polynomial, first we have to combine its like terms:

When performing the calculations, all the third-degree terms of the polynomial have been canceled, therefore the degree of the polynomial is 2.

### Problem 3

Identify the degree of the following polynomial with 2 variables.

**See solution**

To find the degree of each term of the polynomial, we have to add its exponents. So, the monomial with the highest degree is , whose degree is 7 (4+3=7).

Therefore, the degree of the polynomial is 7.

### Problem 4

Find the degree of the following polynomial with four variables.

**See solution**

To find the degree of each monomial of the polynomial, we have to add its exponents. Therefore:

The highest degree of the terms of the polynomial is 10, so it is a polynomial of degree 10.