Polynomials

An expression that can have 

• constants (like 4), 
• variables (like x or y) and 
• exponents (like the 2 in y²), 

that can be combined using 

• addition, 
• subtraction, 
• multiplication and 
• division, 

but:

• no division by a variable.
• a variable's exponents can only be 0,1,2,3,... etc.
• it can't have an infinite number of terms.

A polynomial looks like this:

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 x and y)

exponents (like the 2 in y2), 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.

These are polynomials:

3x

x − 2

−6y² − (79)x

3xyz + 3xy²z − 0.1xz − 200y + 0.5

512v⁵ + 99w⁵

5

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

And these are not polynomials

• 3xy^-2 is not, because the exponent is "-2" (exponents can only be 0,1,2,...)
• 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:

There is also quadrinomial (4 terms) and quintinomial (5 terms), 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: x⁴ − 2x² + x   has three terms, but only one variable (x)

Or two or more variables

Example: xy⁴ − 5x²z   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: x⁴−2x²+x

See how nice and smooth the curve is?

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:

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: 3x² − 7 + 4x³ + x⁶

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

x⁶ + 4x³ + 3x² − 7

You don't have to use Standard Form, but it helps.