Polynomials

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:

that can be combined using addition, subtraction, multiplication and division …

… except …

So:

These are polynomials:

(Yes, even “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

Or one variable

Or two or more variables

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.

source : https://www.mathsisfun.com/algebra/polynomials.html