## Determining the Degree of Polynomials

In this section, we will work with polynomials that have only one variable in each term. The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial that has no variable, just a constant, is a special case. The degree of a constant is \(0\)—it has no variable.

### Definition: Degree of a Polynomial

The **degree of a term** is the exponent of its variable.

The **degree of a constant** is \(0.\)

The **degree of a polynomial** is the highest degree of all its terms.

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

Remember: Any base written without an exponent has an implied exponent of \(1.\)

## Example

Find the degree of the following polynomials:

- \(\phantom{\rule{0.2em}{0ex}}4x\)
- \(\phantom{\rule{0.2em}{0ex}}3{x}^{3}-5x+7\)
- \(\phantom{\rule{0.2em}{0ex}}-11\)
- \(\phantom{\rule{0.2em}{0ex}}-6{x}^{2}+9x-3\)
- \(\phantom{\rule{0.2em}{0ex}}8x+2\)

### Solution

\(4x\) | |

The exponent of \(x\) is one. \(x={x}^{1}\) | The degree is 1. |

\(3{x}^{3}-5x+7\) | |

The highest degree of all the terms is 3. | The degree is 3 |

\(-11\) | |

The degree of a constant is 0. | The degree is 0. |

\(-6{x}^{2}+9x-3\) | |

The highest degree of all the terms is 2. | The degree is 2. |

\(8x+2\) | |

The highest degree of all the terms is 1. | The degree is 1. |

Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in **standard form**. Look back at the polynomials in the example above. Notice that they are all written in standard form. Get in the habit of writing the term with the highest degree first.