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 
Name 
Examples 
1 
Monomial 
3x^{2} 
2 
Binomial 
4x  2 
3 
Trinomial 
2x^{2} + 4x  3 
4 
Poly of 4 terms 
5x^{3} + 2x^{2} + 8x  2x^{5} 
5 
Poly of 5 terms 
4 + 3x^{2} + x^{6}  8x  10x^{2} 
...etc. 
Poly of _ terms 
4x^{2}  5x  11 + ... + 3x + 7x^{5} 
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...
Degree 
Name 
Examples 
0 
Constant 
5 
1 
Linear 
3x + 7 
2 
Quadratic 
2x^{2} + 4x  3 
3 
Cubic 
5x^{3} + 2x 
4 
Quartic 
3x^{4} + 2x^{3}  8 
5 
Quintic 
x^{5}  2x^{3} + 6x  3 
6 
Polynomial of degree 6 
2x^{6} 
...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: 
x^{2} + 4x  8 
(quadratic trinomial) 



Example #2: 
2x^{3}  16x 
(cubic binomial) 
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 x^{2} and x^{3}, 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: x^{2}, x^{4}, x^{6},...) 
Positive: Up, Up 
Negative: Down, Down 


Odd Degree (ex: x, x^{3}, x^{5},...) 
Positive: Down, Up 
Negative: 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.
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