Types of Polynomials
There are two basic methods for classifying polynomials. One is based on the size of the exponents and the other is based on the number of terms.
Definition
The degree of a polynomial with one term is the sum of the exponents of its variables. The degree of a polynomial is equal to the greatest degree of its terms. The degree of a constant, i.e. a term with no variables, is zero.
| Term | Degree |
|---|---|
| $3xy^2 = 3x^1y^2$ | 1 + 2 = 3 |
| $2x = 2x^1$ | 1 |
| $3x^4y = 3x^4y^1$ | 4 + 1 = 5 |
| 2 | 0 |
Let's try an example. Say I asked you for the degree of $3xy^2 + 2x + 3x^4y + 2$. The first thing you would need to do would be to add up the exponents of each of the terms. The chart on the right gives the values for our polynomial. First, notice how I wrote out the 1 exponent on the variables that didn't have an exponent already. We don't usually write out the 1's but they're still there and have to be counted in this process. Once I had an exponent for every variable, I added them up to get the degree of each term. (Remember that the degree of a constant, i.e. a number by itself, is always zero.) The degree of a polynomial is equal to the degree of its biggest term so, in this example, our polynomial's degree must be five.
The second method for categorizing polynomials is based on the number of terms that it has (to give you some more examples to look at, I've added the degrees of the polyomials as well):
| Name | Description | Examples | Degree |
|---|---|---|---|
| monomial | a polynomial with only one term | 3 x 2xyz |
0 1 3 |
| binomial | a polynomial with exactly two terms | 3x + 1 $x^3$ $2xyz + xz$ |
1 1 3 |
| trinomial | a polynomial with exactly three terms | $x+y + 3$ $x^2+x + 3$ $2xyz + xy - 2$ |
1 2 3 |
| quadratic | a second degree polynomial with only one variable | $x^2+x+1$ $x^2-3x + 2$ $x^2-1$ |
2 2 2 |
A quadratic polynomial is actually a special case where we use both the exponents and the number of terms to determine whether or not a polynomial belongs in the category. It may seem strange to single that specific combination of requirements out but, it turns out, that polynomials with those specific traits have a lot of interesting properties and uses - enough to justify giving them their own name.
