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Polynomials

A polynomial looks like this:
polynomial example
example of a polynomial
this one has 3 terms

It can be made of:

circle constants (like 3, -20, or ½)
circle variables (like x and y)
circle exponents (like the 2 in y2) but they can only be 0, 1, 2, 3, ... etc

That can be combined using:

+ - × addition, subtraction and multiplication, ...
 
circle ... but not division! circle

 

Those rules keeps polynomials simple, so they are easy to work with!

Polynomials or Not?

polynomial

These are polynomials:

  • 3x
  • x - 2
  • 3xyz + 3xy2z - 0.1xz - 200y + 0.5

And these are not polynomials

  • 2/(x+2) is not, because dividing is not allowed
  • 3xy-2 is not, because the exponent is "-2" (exponents can only be 0,1,2,...)

But this is allowed:

  • x/2 is allowed, because it is also (½)x (the constant is ½, or 0.5)
  • also 3x/8 for the same reason (the constant is 3/8, or 0.375)

Monomial, Binomial, Trinomial

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

monomial, binomial, trinomial

How do you remember the names? Think cycles!
mono tri bi

(There's also quadrinomial (4 terms) and quintinomial (5 terms), but these are not often used)

Lots and Lots of Terms

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

What is Special About Polynomials?

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

For example we know that:

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

Degree

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

Example:

4x3-x-3 The Degree is 3 (the largest exponent of x)

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