Degree of a Polynomial

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The degree of a polynomial is a very straightforward concept that is really not hard to understand.

In this lesson, you will learn

  • what the degree of a polynomial is
  • how to find the exponent of every term, including terms like 6x and 5
  • how to write a polynomial that has a given number of terms and a given degree
  • how to find the degree when a term has more than one variable

What Is the Degree of a Polynomial?

Table of polynomials and their degrees: 9 has degree 0, x¹ + 4 has degree 1, x² + x¹ + −5 has degree 2, 2x³ + 11x² has degree 3, and x²y³ + 6x⁴ + y¹ has degree 5.
Definition

The degree is the greatest exponent of the terms.

Recall that for y2, y is the base and 2 is the exponent.

What is the degree of 3x5 + x2 − 7?

Show answer

5. The exponents are 5, 2 and 0. The greatest is 5.

Finding the Degree: Two Examples

Example #1: 4x2 + 6x + 5

This polynomial has three terms. The first one is 4x2, the second is 6x, and the third is 5.

The exponent of the first term is 2.

The exponent of the second term is 1 because 6x = 6x1.

The exponent of the third term is 0 because 5 = 5x0.

What? 5x0 = 5?

Well, anything (except 0) with an exponent of 0 is always equal to 1.

Thus, 5x0 = 5 × x0 = 5 × 1 = 5

Since the highest exponent is 2, the degree of 4x2 + 6x + 5 is 2.

Example #2: 2y6 + 1y5 + −3y4 + 7y3 + 9y2 + y + 6

This polynomial has seven terms. The first one is 2y6, the second is 1y5, the third is −3y4, the fourth is 7y3, the fifth is 9y2, the sixth is y, and the seventh is 6.

The exponent of the first term is 6.

The exponent of the second term is 5.

The exponent of the third term is 4.

The exponent of the fourth term is 3.

The exponent of the fifth term is 2.

The exponent of the sixth term is 1 because y = y1.

The exponent of the last term is 0 because 6 = 6y0.

Since the highest exponent is 6, the degree of 2y6 + 1y5 + −3y4 + 7y3 + 9y2 + y + 6 is 6.

Common mistake

Looking only at the first term. Terms are not always written from the greatest exponent to the smallest. In 3x2 + 5x4 − 1, the first term has exponent 2, but the degree is 4.

Worked example: Find the degree of 7 − 2x + x3 − 4x2.
  1. List the terms: 7, −2x, x3 and −4x2.
  2. Write the exponent of each term: 7 = 7x0 has 0, −2x has 1, x3 has 3, and −4x2 has 2.
  3. The greatest exponent is 3, even though x3 is not the first term.

Answer: the degree is 3

What is the degree of the polynomial 9?

Show answer

0. 9 = 9x0, so its only exponent is 0.

Writing a Polynomial From a Description

Write a polynomial for the following descriptions

Remember

A monomial has 1 term, a binomial has 2 terms and a trinomial has 3 terms. Many answers are correct: each answer below is one example.

1) A binomial in z with a degree of 10

Answer: 2z10 − 4

2) A trinomial in c with a degree of 4

Answer: c4 + c2 − 8

3) A binomial in y with a degree of 1

Answer: y + 4

4) A monomial in b with a degree of 3

Answer: b3

Write a trinomial in x with a degree of 5.

Show answer

For example, x5 + 2x − 3. It has 3 terms, and its greatest exponent is 5.

Degree of a Term With More Than One Variable

Rule

To find the degree of a polynomial or monomial with more than one variable for the same term, just add the exponents for each variable to get the degree.

Find the degree of x3y2 + x + 1.

The degree of this polynomial is the degree of the monomial x3y2

Since the degree of x3y2 is 3 + 2 = 5, the degree of x3y2 + x + 1 is 5

What is the degree of 4a2b5 − ab + 3?

Show answer

7. 4a2b5 has degree 2 + 5 = 7, ab has degree 1 + 1 = 2, and 3 has degree 0. The greatest is 7.

Summary

  • The degree of a polynomial is the greatest exponent of its terms.
  • A term like 6x has exponent 1, and a number like 5 has exponent 0.
  • Check every term: the term with the greatest exponent is not always written first.
  • Monomial: 1 term. Binomial: 2 terms. Trinomial: 3 terms.
  • When a term has more than one variable, add their exponents to get the degree of that term.

Test Yourself: 10-Question Quiz

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