All About Polynomials

A main topic in algebra classes is polynomials. There are many subtopics of this topic, including adding, subtracting, multiplying, and dividing.

This page will take you through the basics, including the definition, labeling according to the number of terms, labeling according to the degree, and the end behavior.

Let's begin with the def'n:

Definition: A monomial or a sum of monomials. A monomial is simply a "term."

For more information about terms, see combining like terms.


Labeling According to the Number of Terms

Terms are seperated by + and - signs. Poly's with 1, 2, or 3 terms have specific names, while poly's of 4 or more terms are simply called polynomials of # terms. Take a look at the table below.

# of Terms
1 Monomial 3x2
2 Binomial 4x - 2
3 Trinomial 2x2 + 4x - 3
4 Poly of 4 terms 5x3 + 2x2 + 8x - 2x5
5 Poly of 5 terms 4 + 3x2 + x6 - 8x - 10x2
...etc. Poly of _ terms 4x2 - 5x - 11 + ... + 3x + 7x5

Again, all you are looking for is the number of terms. When there are 1, 2, or 3 the poly is given the special names monomial, binomial, trinomial, respectively. If there are more than 3, then it is simply called a poly of that many terms.

Labeling According to the Degree

The degree of a poly is determined by the exponents. In fact, it is determined by the largest exponent. Take a look at this table...

0 Constant 5
1 Linear 3x + 7
2 Quadratic 2x2 + 4x - 3
3 Cubic 5x3 + 2x
4 Quartic 3x4 + 2x3 - 8
5 Quintic x5 - 2x3 + 6x - 3
6 Polynomial of degree 6 2x6
...etc. Polynomial of degree # (Largest exponent is higher than 6)


Notice that we are no longer interested in the number of terms, but simply the degree of the exponent.

Often polynomials will be referred to by both the number of terms and the degree. Take a look at the two examples below.

Example #1: x2 + 4x - 8 (quadratic trinomial)
Example #2: 2x3 - 16x (cubic binomial)

Standard Form

For simplicity, it is often preferred to put a poly into standard form. This means that the terms are place in descending order (highest to lowest) according to their degree.

End Behavior of the Polynomial

The end behavior of a graph is what happens at the far left and the far right. Two factors determine the end behavior: positive or negative, and whether the degree is even or odd.

For the examples below, we will use x2 and x3, but the end behavior will be the same for any even degree or any odd degree. However, keep in mind that what happens "in the middle" will change...

Even Degree (ex: x2, x4, x6,...)
Positive: Up, Up
Negative: Down, Down
Polynoial End Behavior Up Up Polynomial End Behavior Down Down

Odd Degree (ex: x, x3, x5,...)
Positive: Down, Up
Negative: Up, Down
Polynomial End Behavior Down Up Polynomial End Behavior Up Down


Knowing the general shape and end behavior is an important step towards understanding polys. With a little practice, you will know how to perform all of the operations associated with polynomials. Take a look at how to perform polynomials division.

Return to free algebra help for more help or visit the GradeA homepage.




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