College Algebra Tutorial 6


College Algebra
Tutorial 6: Polynomials


WTAMU > Virtual Math Lab > College Algebra

 

deskLearning Objectives


After completing this tutorial, you should be able to:
  1. Identify a term, coefficient, constant term, and polynomial.
  2. Tell the difference between a monomial, binomial, and trinomial.
  3. Find the degree of a term and polynomial.
  4. Combine like terms.
  5. Add and subtract polynomials.
  6. Multiply any polynomial times any other polynomial.
  7. Use the FOIL method to multiply a binomial times a binomial.
  8. Use special product rules to multiply a binomial squared and a product of a sum and difference of two terms.




desk Introduction


In this tutorial we will be looking at the different components of polynomials.  Then we will move on to adding, subtracting and multiplying them.  Some of these concepts are based on ideas that were covered in earlier tutorials.  A lot of times in math you use previous knowledge to learn new concepts.  The trick is to not reinvent the wheel each time but recognize what you have done before and draw on that knowledge to help you work through the problems. 

 

 

desk Tutorial



Let’s start with defining some words before we get to our polynomial.

  Term
 
A term is a number, variable or the product of a number and variable(s). 

Examples of terms are term, z.


 
 
  Coefficient
 
A coefficient is the numeric factor of your term. 

Here are the coefficients of the terms listed above:
 

Term Coefficient term 3 term 5 term 2 z 1
 
 
  Constant Term
 
A constant term is a term that contains only a number. In other words, there is no variable in a constant term. 

Examples of constant terms are 4, 100, and -5.


 
 
  Standard Form of a Polynomial 

standard form

where n is a non-negative integer. 

Lead coefficient is called the leading coefficient.

constant is a constant.


 
In other words, a polynomial is a finite sum of terms where the exponents on the variables are non-negative integers.  Note that the terms are separated by +'s and -'s.

An example of a polynomial expression is polynomial.


 

Degree of a Term
 
The degree of a term is the sum of the exponents on the variables contained in the term. 

For example, the degree of the term 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 term would be 3 since the only variable exponent that we have is 3.


 
  Degree of the Polynomial
 
The degree of the polynomial is the largest degree of all its terms.

 
 
  Descending Order
 
Note that the standard form of a polynomial that is shown above is written in descending order.  This means that the term that has the highest degree is written first, the term with the next highest degree is written next, and so forth

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.


 
 
  Some Types of Polynomials
  Type Definition Example Monomial A polynomial with one term 5x Binomial A polynomial with two terms 5x - 10 Trinomial A polynomial with three terms trinomial
 
 

notebook Example 1:   Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these: example 1a.

videoView a video of this example


 
Since the degree of the polynomial is the highest degree of all the terms, it looks like the degree is 3. 

Since there are three terms, this is a trinomial.


 
 
 

notebook Example 2:   Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these: example 2a.

videoView a video of this example


 
Since the degree of the polynomial is the highest degree of all the terms, it looks like the degree is 3. 

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.


 
 
 

notebook Example 3:   Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these:   -20.

videoView a video of this example


 
Since the degree of the polynomial is the highest degree of all the terms, it looks like the degree is 0. 

Since there is one term, this is a monomial.


 
 
  Combining Like Terms
 
Recall that like terms are terms that have the exact same variables raised to the exact same exponents.  One example of like terms is liek terms .  Another example is lie terms.

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.


 
  Adding and Subtracting Polynomials
 
Step 1:   Remove the ( ) .

 
If there is only a + sign in front of ( ), then the terms inside of ( ) remain the same when you remove the ( ).

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 ( )).

 

Step 2:  Combine like terms.

 
 
 

notebook Example 4:   Perform the indicated operation and simplify: example 4a.

videoView a video of this example


 
example 4b

*Remove the (  )
*Add like terms together

 
 
 

notebook Example 5:   Perform the indicated operation and simplify: example 5a.

videoView a video of this example


 
example 5b

*Dist. the - through second ( )
*Combine like terms

 
 
  Multiplying Polynomials
 
In general, when multiplying two polynomials together, use the distributive property until every term of one polynomial is multiplied times every term of the other polynomial.  Make sure that you simplify your answer by combining any like terms.

On this page we will look at some of the more common types of polynomials to illustrate this idea.


 
  (Monomial)(Monomial)
 
In this case, there is only one term in each polynomial.  You simply multiply the two terms together.

 
 
 

notebook Example 6:   Find the product example 6a.

videoView a video of this example


 
example 6b
*Mult. like bases add exp.

 
 
  (Monomial)(Polynomial)
 
In this case, there is only one term in one polynomial and more than one term in the other.  You need to distribute the monomial to EVERY term of the other polynomial.

 
 
 

notebook Example 7:   Find the product example 7a.

videoView a video of this example


 
example 7b

*Dist. -5b
*Mult. like bases add exp.


 

(Binomial)(Binomial)
 
In this case, both polynomials have two terms.  You need to distribute both terms of one polynomial times both terms of the other polynomial.

One way to keep track of your distributive property is to use the FOIL method.   Note that this method only works on (Binomial)(Binomial).
 
 

F First terms O Outside terms I Inside terms L Last terms

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.


 
 
 

notebook Example 8:   Find the product example 8a.

videoView a video of this example


 
example 8b

 
 

*Use the FOIL method

*Combine like terms
 


 

Binomial Squared

square
 

Special product rule for 
a binomial squared: 

square

square


 
In other words, when you have a binomial squared, you end up with the first term squared plus (or minus) twice the product of the two terms plus the last term 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 square, you will already be familiar with the product it came from.


 
 
 

notebook Example 9:   Find the product example 9a.

videoView a video of this example


 
example 9b
*square


 

Product of the sum and difference 
of two terms

difference


 
This is another 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 a difference of two squares, you will already be familiar with the product it came from.

 
 
 

notebook Example 10:   Find the product example 10a.

videoView a video of this example


 
example 10b
*difference

 
 
  (Polynomial)(Polynomial)
 
As mentioned above, use the distributive property until every term of one polynomial is multiplied by every term of the other polynomial.  Make sure that you simplify your answer by combining any like terms.

 
 
 

notebook Example 11:   Find the product example 11a.

videoView a video of this example


 
example 11b
*Use Dist. Prop. twice

*Combine like terms

 


 
  Special product rule for 
binomial cubed:

cube

cube


 
In other words, when you have a binomial cubed, you end up with the first term cubed plus (or minus) three times the first term squared times the second term plus three times the first term times the second term squared plus (or minus) the last term cubed.

Any time you have a binomial cubed you can use this shortcut method to find your product.


 

notebook Example 12:   Find the product example 12a.

videoView a video of this example


 
example 12b
*cube

 

 

desk Practice Problems


These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

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.

 

pencil Practice Problems 1a - 1c: Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these.


1a.  -10
(answer/discussion to 1a)
1b. prob 1b
(answer/discussion to 1b)
1c. prob 1c
(answer/discussion to 1c)

 

 

pencil Practice Problems 2a - 2e: Perform the indicated operation.

 

2a. prob 2a
(answer/discussion to 2a)
2b. prob 2b
(answer/discussion to 2b)

 
2c. prob 2c
(answer/discussion to 2c)
2d. prob 2d
(answer/discussion to 2d)

 
2e. prob 2e
(answer/discussion to 2e)

 

 

desk Need Extra Help on these Topics?


 
 
The following are webpages that can assist you in the topics that were covered on this page:
 
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut25_poly.htm
This webpage goes over the basic terminology of polynomials as well as how to add and subtract them.

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.


 

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Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.
Last revised on Dec. 13, 2009 by Kim Seward.
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