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Introduction to Polynomials

Introduction

Intro_to_Polynomials_visuals_1An expression containing several different variable terms is called a polynomial.  Notice that each term in the polynomial to the right contains a different power of x.

 

Most of the polynomials that you will encounter in the next few lessons will contain terms that have the same variable but to different powers.  Polynomials containing two or more different variables are much more complex and are not generally covered until a much later mathematics course.

Intro_to_Polynomials_visuals_2         Intro_to_Polynomials_visuals_3

 

Lesson

In this lesson, vocabulary is especially important, so take your time and make sure you can differentiate between the prefixes mono, bi, tri, and poly.


Mono means “one” and can be found in words like monopoly (one controlling interest), monogamous (having one spouse), monorail (train riding on a single rail), and even monarchy (government with one ruler).”

 

Intro_to_Polynomials_visuals_4

 

 

A monomial contains one term  

 

 

 

 

Bi means “two” and can be found in words like bicycle (cycle with two wheels), bipolar (having two separate attitudes or personalities), bilateral (two sided).


Intro_to_Polynomials_visuals_5

 

 

A binomial contains two terms  

 

 

 

 

Tri means “three” and can be found in words like tripod (having three legs), tricycle (cycle with three wheels), triumvirate (rule by three individuals), and triangle (three sided polygon).


Intro_to_Polynomials_visuals_6

 

 

A trinomial contains three terms  

 

 

 

 

The prefix poly means “many” and can be found in words like polygon (having many sides), polytheism (belief in multiple gods), polygraph (machine that detects truthfulness using several physiological indices), polygamy (having multiple spouses).

Intro_to_Polynomials_visuals_7

 

 

Polynomials are expressions that contain more than one term.  Binomials and trinomials are polynomials.  However, polynomials are generally thought of as being larger than simply a binomial or trinomial.  Any expression with 4, 5, 6, or more terms is also called a polynomial.   

 

 

 


Intro_to_Polynomials_visuals_8

A number out in front of a term is called a coefficient, while the small raised number is called the exponent .  

 

 

Since polynomials contain multiple terms, the rules of mathematics allow the terms to be arranged in any order.  When confronted with a polynomial, it is common practice to rearrange the terms so that the exponents are listed in descending order.

Intro_to_Polynomials_visuals_9

 

The word standard form describes a polynomial whose terms are listed in descending order.

Consider how you would put the polynomial into standard form.


Intro_to_Polynomials_visuals_10                                  Intro_to_Polynomials_visuals_10a

 

 

Intro_to_Polynomials_visuals_11                Intro_to_Polynomials_visuals_11a



 

*Notice how the red circle above (in the 2nd problem) includes the entire (- 18b).  Remember that every minus sign goes with the term that comes immediately after it.  When working with integers, you learned to “add the opposite” to get rid of a minus sign.  Therefore, - 18b is the same as + (-18b).  Now that you are an algebra student, you may be expected to remember this fact in your head and may not be required to show this work on paper.  If you would like a review on this topic, feel free to visit the lesson on adding integers.

 

 

A polynomial’s degree describes the largest exponent in the polynomial.  The largest variable in the first problem above is a 4 (from 5a4) so the polynomial is to the 4th degree.  In the second problem above, the largest variable is a 7 (from 15b7) so the polynomial is to the 7th degree.

 

 

One final phrase you should be familiar with is the phrase leading coefficient.  This means the coefficient of the term that has the largest exponent.  An easier way to remember this is to make sure the expression is in standard form… then the leading coefficient refers to the coefficient of the first term.


Intro_to_Polynomials_visuals_12

 

In the diagram to the left, the polynomial is written in two different ways.  Notice that it is much easier to find the leading coefficient of 15 in the lower example after the polynomial is changed into standard form.

 

 

Try It

1)  What is the difference between a binomial and a trinomial?

2)  Can a binomial also be a polynomial?

3)  Is the polynomial 3r4 – 8r3 + 14r2 – 9r + 6 written in standard form?

4)  What is the leading coefficient of 6y2 + 11y3 + 14y + 9?

5)  What is the leading coefficient of z5 + 3z3 + 21z + 15?

Change each polynomial into standard form.  Then state the degree of each polynomial. 

6)  4a3 +  2a5 + a4 + a2 – 18a

7)  5b4 + 3b3 – 6b2 + 21b + 40

8)  5c11 + 17c5 + 3c8 + 2c3 + 4c9

 

Scroll down for answers...

 

 

 

 

 

 

 

Answers:

1)  A binomial is an algebraic expression that contains 2 terms.  A trinomial is an algebraic expression that contains 3 terms.

 

2)  Yes.  A binomial contains two terms and a polynomial contains 2 or more terms, so every binomial is also a polynomial.

 

3)  Yes.  The exponents are in descending order, so the polynomial is in standard form.

 

4)  11.  The leading coefficient comes from term with the largest exponent, which is 11y3.

 

5)  1.  The term with the largest exponent is z5.  Since there is no number in front of the z, it is assumed to be a 1.

 

 

Change each polynomial into standard form.  Then state the degree of each polynomial. 

6)  2a5 + a4 + 4a3 + a2 – 18a

7)  5b4 + 3b3 – 6b2 + 21b + 40   (This was already in standard form)

8)  5c11 + 4c9 + 3c8 + 17c5 + 2c3

 

 

 

 

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