Learning Objectives
Introduction
Tutorial
Examples of terms are , z.
Here are the coefficients of the terms listed above:
Examples of constant terms are 4, 100, and -5.
where n is a non-negative integer.
is called the leading coefficient.
is a constant.
An example of a polynomial expression is .
Degree of a TermFor example, the degree of the term would be 1 + 1 = 2. The exponent on a is 1 and on b is 1 and the sum of the exponents is 2.
The degree of the term would be 3 since the only variable exponent that we have is 3.
Also note that a polynomial can be “missing” terms. For example, the polynomial written above starts with a degree of 5, but notice there is not a term that has an exponent of 4. That means the coefficient on it is 0, so we do not write it.
Example 1: Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these: .
Since there are three terms, this is a trinomial.
Example 2: Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these: .
Make sure that you don’t fall
into the trap
of thinking it is always the degree of the first term. This
polynomial
is not written in standard form (descending order). So we had to
actually go to the second term to get the highest degree.
Since there are two terms, this is a binomial.
Example 3: Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these: -20.
Since there is one term, this is a monomial.
You can only combine terms that are like terms. You can think of it as the reverse of the distributive property.
It is like counting apples and oranges. You just count up how many variables you have the same and write the number in front of the common variable part.
If there is a - in front of the ( ), then distribute it by multiplying every term in the ( ) by a -1 (or you can think of it as negating every term in the ( )).
Example 4: Perform the indicated operation and simplify: .
Example 5: Perform the indicated operation and simplify: .
On this page we will look at some of the more common types of polynomials to illustrate this idea.
Example 6: Find the product .
Example 7: Find the product .
One way to keep track of your distributive property
is to use the FOIL method. Note that this method
only works
on (Binomial)(Binomial).
This is a fancy way of saying to take every term of the first binomial times every term of the second binomial. In other words, do the distributive property for every term in the first binomial.
Example 8: Find the product .
*Use the FOIL method
*Combine like terms
Special product rule for
a binomial squared:
Any time you have a binomial squared you can use this shortcut method to find your product.
This is a special products rule. It would be perfectly ok to use the foil method on this to find the product. The reason we are showing you this form is that when you get to factoring, you will have to reverse your steps. So when you see , you will already be familiar with the product it came from.
Example 9: Find the product .
Example 10: Find the product .
Example 11: Find the product .
*Combine like terms
Any time you have a binomial cubed you can use this shortcut method to find your product.
Example 12: Find the product .
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1c: Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these.
Practice Problems 2a - 2e: Perform the indicated operation.
Need Extra Help on these Topics?
http://www.purplemath.com/modules/polydefs.htm
This webpage helps you with the different parts of a polynomial.
http://www.purplemath.com/modules/polyadd.htm
This webpage helps you with adding and subtracting polynomials.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut26_multpoly.htm
This webpage goes over multiplying polynomials.
http://www.algebrahelp.com/lessons/simplifying/distribution/
This website helps with the distributive property.
http://www.algebrahelp.com/lessons/simplifying/foilmethod/
This website helps with the FOIL method and (polynomial)(polynomial).
http://www.purplemath.com/modules/polymult.htm
This webpage helps with multiplying polynomials.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.
Last revised on Dec. 13, 2009 by Kim Seward.
All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.