# 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"

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 |

**addition, subtraction, multiplication and division**...

... except ...

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

A polynomial can have constants, variables and exponents,

but never division by a variable.

but never division by a variable.

## Polynomial or Not?

**are**

**polynomials:**

**3x****x - 2****-6y**^{2}- (^{7}/_{9})x**3xyz + 3xy**^{2}z - 0,1xz - 200y + 0,5**512v**+^{5}**99w**^{5}**5**

**one term is allowed**, and it can even be just a constant!)

And 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"

**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.

but those names are not often used.

## Can Have Lots and Lots of Terms

Polynomials can have as many terms as needed,**but not an infinite number of terms**.

## 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

Or two or more variables^{4}-2x^{2}+x has three terms, but only one variable (x)
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

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 smooth the curve is? |

## Degree

The**degree**of a polynomial with only one variable is the

**largest exponent**of that variable.

### Example:

The Degree is 3 (the largest exponent of x) |

## 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}

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

^{3}

^{6}

**x**+ 4

^{6}**x**+ 3

^{3}**x**- 7

^{2}
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