# Polynomial

There are several conditions that need to be met in order to determine whether a mathematical expression is a **polynomial**. First, polynomials consist of *constants* and *variables*. These constants and variables make separable parts of a __polynomial__ called *terms* and polynomials are made up of a finite number of these terms. The terms are separated by the symbols of mathematical operations (+,-,*), which brings us to the second condition.

The only mathematical operations which can be used in polynomials are addition, subtraction and multiplication. Division, in which the variable is a part of the denominator, does not produce a polynomial. If the denominator is a constant, a polynomial can be produced.

The third condition and the defining characteristic of polynomials are *powers* (or exponents). We covered exponents in the article called Squares and square roots, so if you are not sure what they are, feel free to refresh your memory. Each term in a polynomial has to have its *non-negative integer exponent*. That means that the value of the exponent for each term can be any integer between (and including) zero and any other finite value.

Terms and polynomials have their *degrees or orders*. The order of a term is determined by the sum of the exponents in that term. The order of a polynomial is the largest one among the orders of the terms that make the polynomial. That means that, for example, polynomials of the fourth order would look somewhat like this:

a*x^{4} + b*x^{3} + c*x^{2} + d*x + e

Or, if they have two variables, like this:

A*x^{2}*y^{2} + b*x*y + c

We can name polynomials according to their degrees (or orders).

Degree |
Name |

0 |
constant |

1 |
linear |

2 |
quadratic |

3 |
cubic |

4 |
quartic (biquadratic) |

5 |
quintic |

6 |
sextic (hexic) |

7 |
septic (heptic) |

8 |
octic |

9 |
nonic |

10 |
decic |

100 |
hectic |

These are the degrees which have special names. For others you can always use the standard nomenclature – eleventh degree, 57^{th} order and such.

Another important thing to know is that polynomials can have any number of terms as long as the number of terms is *finite*, no matter what their order is. Depending on that number, we name them:

Number of non-zero terms |
Name |

0 |
zero polynomial |

1 |
monomial |

2 |
binomial |

3 |
trinomial |

All other polynomials are simply called – polynomials.

What is interesting about the* zero polynomial* is that its order is intentionally left explicitly undefined or is defined to be negative, usually with the values of -1 or negative infinity. Also, it is the only one with an infinite number of roots.

Polynomials have several basic properties, a few of which we are going to mention here. The first one is that the sum of polynomials is always a polynomial. Also the product of polynomials is itself a polynomial. The same thing goes for the composition of two polynomials.

## Simplifying a polynomial

Solving polynomials of the third order or greater requires some advanced mathematical knowledge and we will show you how to do it when the time comes. But for now we will stick to the simplification of polynomials.

There are a few things you should have in mind when simplifying polynomials. The most important thing to remember is that addition and subtraction can be performed only with terms that have the same variables and of the same order! That is a general rule in mathematics, but it is particularly observable in this case. Also, all rules about the order of operations apply.

If after all this polynomials remind you of equations – you are right. In fact, polynomials perform as any other equations and you can treat them as such. But the reason we are mentioning them separately from equations is that they are the gateway to more complex mathematics and it is of vital importance that you understand them and know how to deal with them properly.

If you wish to practice what you learned and try to simplify a polynomial or two, please feel free to use the math worksheets below.