A polynomial is basically a string of mathematical clumps (called terms) all added together. Each individual clump usually consists of one or more variables raised to exponential powers, usually with a coefficient attached. Polynomials can be as simple as the expression 4x, or as complicated as the expression 4x3 + 3x2 - 9x + 6.
Polynomials are usually written in standard form, which means that the terms are listed in order from the largest exponential value to the term with the smallest exponent. Because the term containing the variable raised to the highest power is listed first in standard form, its coefficient is called the leading coefficient. A polynomial not containing a variable is called the constant.
Talk the Talk
A polynomial consists of the sum of distinct algebraic clumps (called terms), each of which consists of a number, one or more variables raised to an exponent, or both. The largest exponent in the polynomial is called the degree, and the coefficient of the variable raised to that exponent is called the leading coefficient. The constant in a polynomial has no variable written next to it.
For example, if you were to write the polynomial 2x3- 7x5 + 8x + 1 in standard form, it would look like this: -7x5 + 2x3 + 8x + 1. (Note that each term's variable has a lower power than the term to its immediate left.) The degree of this polynomial is 5, its leading coefficient is -7, and the constant is 1.
Technically, the constant in a polynomial does have a variable attached to it, but the variable is raised to the 0 power. For example, you could rewrite the simple polynomial 2x + 1 as 2x + 1x0, but since x0 = 1 (and anything multiplied by 1 equals itself), there's no reason to write x0 at the end of the polynomial.
Because there are so many different kinds of polynomials (52 flavors at last check, including pistachio), there are two techniques that are used to classify them, one based on the number of terms a polynomial contains (see Table 10.1), and one based on the degree of the polynomial (see Table 10.2).
Table 10.1 Classifying a Polynomial Based on the Number of Its Terms
|Number of Terms||Classification||Example|
|2||binomial||3x3 - 7x2|
|3||trinomial||2x2 + 5x - 1|
Notice that there are only special classifications for polynomials according to the number of their terms if that number is three or less. Polynomials with four or more terms are either classified according to degree or just described with the ultra-generic (and not very helpful) label "polynomial." (It's just as specific as labeling you a "human being.")
Table 10.2 Classifying a Polynomial Based on Its Degree
|0||constant||2x0 or 2|
|1||linear||6x1 + 9 or 6x + 9|
|2||quadratic||4x2 - 25x + 6|
|3||cubic||x3 - 1|
|4||quartic||2x4 - 3x2 + x - 8|
|5||quintic||3x5 - 7x3 - 2|
If you're asked to classify a polynomial like 3x3y2 - 4xy3 + 6x (which contains more than one kind of variable in some or all of its terms) according to its degree, add the exponents in each term together. The highest total will be the degree. In 3x3y2- 4xy3 + 6x, the degree is 5, since the highest exponent total comes from the first term, and 3 + 2 = 5.
There are more degree classifications for polynomials, but those listed in Table 10.2 are by far the most commonly used.
When classifying a polynomial, you don't have to choose one method or the other. In fact, if you classify the polynomial both ways at once, whenever possible, you paint a more descriptive picture of it.
You've Got Problems
Problem 1: Classify the following polynomials:
(a) 4x3 + 2
Example 1: Classify the following polynomials.
- (a) 3 - 4x - 6x2
- Solution: This polynomial has three terms, so it's a trinomial. Furthermore, its degree is 2, which makes it quadratic. So, all together, it's a quadratic trinomial. When you use both classifications at once, write the degree classifier first since it's an adjective ("trinomial quadratic" just doesn't sound right).
- (b) 13
- Solution: There's only one term, and it has no variable written explicitly; therefore, this is the same thing as 13x0. This expression is best classified as a constant monomial.
Excerpted from The Complete Idiot's Guide to Algebra 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.