Degree of a polynomial

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.

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:

x^5+2x^4+6x^2-3

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:

 P(x) = 4

  • Example of a first degree polynomial:

 P(x) = 3x+2

  • Example of a second degree polynomial:

 P(x) = x^2+7x-4

  • Example of a third degree polynomial:

 P(x) = 2x^3+5x^2-9

  • Example of a fourth degree polynomial:

 P(x) = 6x^4+3x^2-7x+1

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 5x4y3 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:

 P(x,y,z) = 3x^5y^4 + 6x^3y^2z - 2y^6z^2

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.

 \text{Degree of } P(x,y,z) = 9

Names of polynomials by degree

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

Degree of the polynomialNameExample
Polynomial of degree 0Constant (or zero polynomial)5
Polynomial of degree 1Linear polynomialx+2
Polynomial of degree 2Quadratic polynomialx2+3x+1
Polynomial of degree 3Cubic polynomialx3-4x+2
Polynomial of degree 4Quartic polynomial5x4+x2+9
Polynomial of degree 5Quintic polynomialx5-2x3+3x
Polynomial of degree 6Sextic polynomial2x6+8x4-x
Polynomial of degree 7Septic polynomial-3x7+4x4+3x
Polynomial of degree 8Octic polynomialx8-9x2+6
Polynomial of degree 9Nonic polynomialx9+7x5-x2
Polynomial of degree 10Decic polynomial8x10+3x8-4x5-6x2

Practice problems on finding the degree of a polynomial

Problem 1

Find the degree of the following polynomial:

x^4+3x^2+10x-5

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?

5x^3+4x^2-x^3+5x-4x^3+1+3x^2

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

(5x^3-x^3-4x^3)+(4x^2+3x^2)+5x+1

7x^2+5x+1

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.

7x^2y^3-2x^4y^3-3x^5y+x^6

To find the degree of each term of the polynomial, we have to add its exponents. So, the monomial with the highest degree is -2x^4y^3, 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.

3a^3b^2c+5a^5cd^2-2a^2bc^4d^3

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

\text{Degree of }3a^3b^2 = 3+2=5

\text{Degree of }5a^5cd^2 = 5+1+2=8

\text{Degree of }-2a^2bc^4d^3 =2+1+4+3=10

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

 

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