# Polynomials

A polynomial looks like this:

example of a polynomial this one has 3 terms |

**Polynomial** comes from *poly-* (meaning "many") and *-nomial* (in this case meaning "term") ... so it says "many terms"

Polynomials with one variable make nice smooth curves:

A polynomial can have:

constants (like 3, −20, or ½) |

variables (like and x)y |

exponents (like the 2 in y^{2}), but only 0, 1, 2, 3, ... etc are allowed |

that can be combined using **addition, subtraction, multiplication and division** ...

... except ...

... not division by a variable (so something like 2/x is right out) |

So:

A polynomial can have constants, variables and exponents,

but never division by a variable.

Also they can have one or more terms, but not an infinite number of terms.

## Polynomial or Not?

These **are** polynomials:

**3x****x − 2****−6y**^{2}− (\frac{7}{9})x**3xyz + 3xy**^{2}z − 0.1xz − 200y + 0.5**512v**+^{5}**99w**^{5}**5**

(Yes, "5" is a polynomial, **one term is allowed**, and it can be just a constant!)

These are **not** polynomials

**3xy**is not, because the exponent is "-2" (exponents can only be 0,1,2,...)^{-2}**2/(x+2)**is not, because dividing by a variable is not allowed**1/x**is not either**√x**is not, because the exponent is "½" (see fractional exponents)

**But** these ** are** allowed:

**x/2****is allowed**, because you can divide by a constant- also
**3x/8**for the same reason **√2**is allowed, because it is a constant (= 1.4142...etc)

## Monomial, Binomial, Trinomial

There are special names for polynomials with 1, 2 or 3 terms:

**How do you remember the names? Think cycles!
**

*There is also quadrinomial (4 terms) and quintinomial (5 terms),
but those names are not often used.*

## Variables

Polynomials can have no variable at all

Example: 21 is a polynomial. It has just one term, which is a constant.

Or one variable

Example: x^{4 }− 2x^{2 }+ x has three terms, but only one variable (x)

Or two or more variables

Example: xy^{4 }− 5x^{2}z has two terms, and three variables (x, y and z)

## What is Special About Polynomials?

Because of the strict definition, polynomials are **easy to work with**.

For example we know that:

- If you add polynomials you get a polynomial
- If you multiply polynomials you get a polynomial

So you can do lots of additions and multiplications, and still have a polynomial as the result.

Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines.

### Example: x^{4}−2x^{2}+x

See how nice and |

You can also divide polynomials (but the result may not be a polynomial).

## Degree

The **degree** of a polynomial with only one variable is the **largest exponent** of that variable.

### Example:

4x^{3} − x + 2 |
The Degree is 3 (the largest exponent of x) |

For more complicated cases, read Degree (of an Expression).

## Standard Form

The Standard Form for writing a polynomial is to put the terms with the highest degree first.

### Example: Put this in Standard Form: 3**x**^{2} − 7 + 4**x**^{3} + **x**^{6}

^{2}

^{3}

^{6}

The highest degree is 6, so that goes first, then 3, 2 and then the constant last:

**x ^{6}** + 4

**x**+ 3

^{3}**x**− 7

^{2}You **don't have to** use Standard Form, but it helps.