- Topics At a Glance
- Exponents
- Negative Exponents
- Fractional Exponents
- Irrational Exponents
- Variables as Exponents
**Defining Polynomials****Degrees of a Polynomial**- Multivariable Polynomials
- Degrees of Multivariable Polynomials
- Special Kinds of Polynomials
- Evaluating Polynomials
- Roots of a Polynomial
- Combining Polynomials
- Multiplying Polynomials
- Multiplication of a Monomial and a Polynomial
- Multiplication of Two Binomials
- Special Cases of Binomial Multiplication
- General Multiplication of Polynomials
- We'll Divide Polynomials Later!
- Factoring Polynomials
- The Greatest Common Factor
- Recognizing Products
- Trial and Error
- Factoring by Grouping
- Summary
- Introduction to Polynomial Equations
- Solving Polynomial Equations
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

Each term in a polynomial has what is called a **degree**, or a value based on the exponents attached to its variable. The degree of 9*x*^{2} is 2, for example. You may be unfamiliar with a degree of 2 unless you have ever been to Fairbanks, Alaska in the middle of January.

We usually write the terms of a polynomial in **descending order** (greatest to least) according to the degree of each term. Exactly like a shuttle launch countdown. For example, we would write 2*x*^{2} + *x* instead of* x* + 2*x*^{2}.

Term | Degree |
---|---|

3x^{2} | 2 |

5x | 1 |

65x^{67} | 67 |

7 | 0 |

The degree of the constant 7 is zero, since 7 = 7*x*. The degree of any other non-zero constant is also zero. However, the degree of the term 0 is undefined. You might want to get on that, Webster's.

Since 0 = 0*x *= 0*x*^{1 }= 0*x*^{2 }= 0*x*^{3} (all of these are the same as 0), any degree would work, and there's no obvious way to decide which one to pick. Mathematicians hate not knowing what to pick, so that's why they say it's "undefined." You should see them in a grocery store looking over grapefruit to see which is the most ripe. "Undefined...undefined...undefined...undefined..."

There is also a degree of each polynomial. You can figure out the **degree of a polynomial** if you haven't forgotten which numbers are bigger than each other. If all else fails, count them off on your fingers and hope you never run into anything bigger than 10.

Memorize this: the degree of a polynomial is the largest degree of any one term in the polynomial. While we usually write polynomials with the largest degree term first, it's a good idea to look at the degrees of all the terms, in case some impish degree sprite came along and mixed them up to make our lives miserable.

Polynomial | Degree |
---|---|

5x^{3} + 6x + 9 | 3 |

x^{23}x^{6} + 4x -2 | 23 |

4x^{6}+3x^{5}+2x^{3} | 6 |

7 | 0 |

4x+5x^{2} | 2 |

The degree 0 term of a polynomial is also called the **constant term** of the polynomialâ€”the number sitting all by itself, usually at the end of the polynomial. Who knows, maybe it couldn't find its deodorant this morning.

If a polynomial doesn't seem to have a constant term, as in 3*x*^{2} + 4*x*, we say its constant term is 0 because we can write "+ 0" at the end of any expression without changing the value of the expression. If you're ever asked to pay $10 + 0 for something, remember that it doesn't cost you anything extra before you decide to get all in a huff about it.

Example 1

In the polynomial 5 |

Example 2

The constant term of the polynomial |

Example 3

The polynomial 5 |

Exercise 1

Find the degree and constant term of the polynomial *x*^{4} + 7*x*^{3} - 2*x*.

Exercise 2

Find the degree and constant term of the polynomial 3*x*^{20} - *x*^{10} + *x*^{9} - 5.

Exercise 3

Find the degree and constant term of the polynomial *x*^{15} + 8