By: Ayman M. Chris G. Sara G. Marisa U.

Polynomials


One of the most important concepts for algerbra throughout mathematics and science



A Polynomial is an algebraic expression made up of terms such as variables, constants, exponents and coefficients.


The power in math comes from variables (letters) not numbers.

Page Contents

Common Terminology
Monomials
Trinomials
Degree of a Polynomial
Ascending Order
Descending Order
Polynomials
Binomials
Degree of a Term
Collecting Like Terms
Use of Polynomial
Beyond the Classroom
Here is some terminology used when working with polynomials:

Variable: is a letter that represents one or more numbers; Usually represented by "x" or any letter in the alphabet

Term: A term is made up of two parts * Terms are Separated by + or a - sign.
1) Numeric Co-effcient (a number) 5x^2
2) Literal Co-effcient (a letter)5x^2

Algebraic Expression: expressions that include numbers and variables

Like Terms: Terms which have the same variable and power. Like terms can be combined using addition or subtraction.
example: 6x²+2x² these terms are like terms because they both have a degree of 2.

Coefficient:
if a term includes a variable, the numerical factor is the coefficient; the number in front of a variable

Constant Term: A term or expression with no variables. Just a number on it's own.


Expression: a mathematical phrase consisting of variables and numbers. An expression does not have an equal sign.

By now, you should be pretty familiar with variables and exponents, and you may have dealt with expressions like 3x4 or 6x. Polynomials are sums of these "variables and exponents" expressions. A polynomial is simply the sum of monomials. Each piece of the polynomial, the part that is being added, is called a "term".The terms are arranged in either ascending or descending order according to their powers. Polynomial terms have variables which are raised to whole-number exponents (or else the terms are just plain numbers); there are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions.
Here are some examples:

6x –2
This is NOT
a polynomial term...

...because the variable has a negative exponent.
1/x2
This is NOT
a polynomial term...

...because the variable is in the denominator.
sqrt(x)
This is NOT
a polynomial term...

...because the variable is inside a radical.
4x2
This IS a polynomial term...
...because it obeys all the rules.

Here is a typical polynomial:
terms
terms

Notice the exponents on the terms. The first term has an exponent of 2; the second term has an "understood" exponent of 1 we no its there we just do not need to write it ; and the last term doesn't have any variable at all. Polynomials are usually written this way, with the terms written in "decreasing" order; that is, with the largest exponent first, the next highest next, and so forth, until you get down to the plain old number.
Any term that doesn't have a variable in it is called a "constant" term because, no matter what value you may put in for the variable
x, that constant term will never change. In the picture above, no matter what x might be, 7 will always be just 7.
The first term in the polynomial, when it is written in decreasing order, is also the term with the biggest exponent, and is called the "leading term".
The exponent on a term tells you the "degree" of the term. For instance, the leading term in the above polynomial is a "second-degree term" or "a term of degree two". The second term is a "first degree" term. The degree of the leading term tells you the degree of the whole polynomial; the polynomial above is a "second-degree polynomial".

Here are a couple more examples:

  • Give the degree of the following polynomial: 2x5 – 5x3 – 10x + 9
This polynomial has four terms, including a fifth-degree term, a third-degree term, a first-degree term, and a constant term.
This is a fifth-degree polynomial.
  • Give the degree of the following polynomial: 7x4 + 6x2 + x
This polynomial has three terms, including a fourth-degree term, a second-degree term, and a first-degree term. There is no constant term.
This is a fourth-degree polynomial.


Monomials, Binomials, Trinomials

Monomial, binomial and trinomial are specials names given to polynomials with a certain number of terms.
Monomials
Binomials
Trinomials
An expression containing one term is called a monomial.
To remember...Think Monocycle!

A monomial: 2x
Not a monomial: 2x-1
An expression containing two terms is called a binomial
To remember...Think Bicycle!

A binomial: 8x-3
Not a binomial: 8x-3+10
An expression containing three terms is called a trinomial.
To remember...Think Tricycle!

A trinomial: 10x-5+4
Not a trinomial: 10x-5
We name polynomials based on the number of terms.
Polynomial Names
2x
One Term Monomial
8x-3
Two Terms Binomial
10x-5+4
Three Terms Trinomial

When a polynomial contains more than three terms we simply call it a polynomial where the prefix 'poly' means 'many'.

When trying to figure out the difference between them, imagine the signs as barriers.
Ex. 2x-3, now imagine this 2x
/3. Since there are two parts to this equation that means that it is a binomial.


Polynomial

Any polynomial with 4 or more terms is called a polynomial.


Degree of a Term

Remember that a polynomial with one term is called a monomial.

The degree of a term/monomial is the sum of the exponents of all of its variables.

To find the degree, add all the exponents of all the variables together. A variable without an exponent has the exponent one. The sum of all the variables is the degree of the term.

Remember that if a variable has no exponent we automatically classify it as having a 1 as the exponent.We do not put the 1 there because that is bad math but we know it is there.

In the first line of the chart, the only variable has the exponent 3, therefore the degree is 3.
In the second line of the chart, x has the exponent 2, y has the exponent 3 and z has the exponent 5. The three exponents are added together, 2+3+5 to get a degree of 8.
In the third line of the chart, there are 4 variables, all with the exponent one. 4X1=4, therefore its degree is 4.




Degree of a Polynomial
The degree of a polynomial in one variable is the highest power of the variable in any one term.

The degree of a polynomial in two or more variables is the greatest sum of the exponents in any one term.

To find the degree, look at each term as an individual term. Find the degree of each term. Whichever term has the highest individual degree is the degree for the whole polynomial.

In the first line, the first term in the equation has a greater degree than the second term, so the first term's degree of 3 is used as the overall degree of the polynomial.
In the second line, the first term has a greater expression than both the second and third terms, so the first term's degree of 4 is used as the overall degree of the polynomial.
In the third line, the second term appears to have a greater degree, but when you add up all the exponents in the first term (1+1+1+1) the degree is 4, making it greater than the second term and the overall degree of the polynomial.


Ascending Order
http://www.adrianbruce.com/maths/posters/ascending_order/ascending_order_sm.gif
http://www.adrianbruce.com/maths/posters/ascending_order/ascending_order_sm.gif
http://www.adrianbruce.com/maths/posters/descending_order/descending_order_poster_sm.jpg
http://www.adrianbruce.com/maths/posters/descending_order/descending_order_poster_sm.jpg

Ascending order is when numbers are arranged from the smallest to the largest number, so they are pretty much rearranged in order from smallest to biggest.
E.g.1, 3, 5, 6 and 8 are arranged in ascending order.
1 being the smallest number and 8 being the largest number.

You can also put numbers with exponents into Ascending order ( and descending order, too ). All you have to do is
-Take the number and multiply it by itself, how many times the exponent number is.
-Once you have the total, put all the totals in Ascending Order.
(Smallest to greatest) .
And that is how you put numbers with exponents into Ascending Order
.

Note:#(^)#.... ^ represents the phrase "to the power of".
Used in Context: 4^6 means to the power of 6 or to the exponent 6.


This is an example *

4^8, 3^2, 7^5, 2^4



Solving:


  • 4^8 = 4x4x4x4x4x4x4x4=65536
  • 7^5 = 7x7x7x7x7=16807
  • 2^4 = 2x2x2x2=16
  • 3^2 = 3x3=9

The numbers arranged in Ascending Order are


3^2
, 2^4, 7^5, 4^8

To Remember, Think of an upwards staircase.

However!!! If a question says "Put the polynomials in ascending order of the degree of "x", find only the degree of the "x" variables, ignoring the exponents of all other variables.

Descending Order

Descending Order is when numbers are arranged from the largest number to the smallest number, so they are rearranged in order from the biggest number to the smallest.

E.g. 12, 8, 5, 2, and 1 are arranged in Descending order.
12 being the biggest, and 1 being the smallest.


Like shown above, you can also put numbers with exponents into Descending Order. You have to follow the same rules listed above, and the answers will look kind of the same, but just instead of smallest to greatest, it will be greatest to smallest.

Here is an example of what a question would look like, and what an answer should look like
.

This is an Example
3^3, 2^5, 2^4


Solving



  • 3^3 =3x3x3=27
  • 2^5 =2x2x2x2x2=32
  • 2^4 =2x2x2x2=16


The numbers arranged in Descending Order are:
2^5, 3^3, 2^4

Another example:



  • 8^3=8x8x8=512
  • 2^4=2x2x2x2=16
  • 6^3=6x6x6=216

The numbers arranged in Descending Order are:
8^3, 6^3, 2^4
To collect Like Terms they have to be the same variable and be raised to the same exponent.

Example; 2x, 3x, 4x - these are like terms because they have the same variable.

Example; 7³, 8³ are like terms because they are raised to the same exponent.

You can add
like terms together to make one term:
Example: 7x + x = 8x
Example: 5y^2 + 6y^5 = 11y^7

When putting polynomials in ascending and descending order, you must first find the degree of each polynomial. Refer to Degree of a Term.
After finding their degree, put them in the according order based on their degree.

Example of Ascending and Descending Order Using Polynomials

Let's use these polynomials: 5yx^3 y^2zx^6 8x^2
First, identify the degrees
y^5x^3 = degree of 8
y^2zx^6 = degree of 9
8x^2 = degree of 2

Then put them in ascending order: (REMEMBER: ASCENDING MEANS SMALLEST TO BIGGEST)
8x^2
y^5x^3
y^2zx^6

Then, put them in descending order (REMEMBER: DESCENDING MEANS BIGGEST TO SMALLEST)
y^2zx^6
y^5x^3

8x^2




If you have an equation like; z+4x-9z-2x=-z-9z+4x-2x




you would bring all the X terms to one side of the equal sign




you would bring all the X terms to one side of the equal sign




and all Z terms to the other side of the equal sign




You now have: 4x-2x-4x+2x=9z-9z+z-z -in this case all the numbers cross each other out




so the answer is simply zero.

Unlike Terms

Unlike terms are really easy

They are the exact opposite of like terms

8x, 8z- these are unlike terms because the have a different variable

9, 9 - these are unlike terms because they are raised to different exponents


When a term contains both a number and a variable part, the number part is called the "coefficient". The coefficient on the leading term is called the "leading" coefficient.
terms
terms

In the above example, the coefficient of the leading term is 4; the coefficient of the second term is 3; the constant term doesn't have a coefficient. Copyright © Elizabeth Stape006-2008 All Rights Reserved
The "poly" in "polynomial" means "many". I suppose, technically, the term "polynomial" should only refer to sums of many terms, but the term is used to refer to anything from one term to the sum of a zillion terms. However, the shorter polynomials do have their own names:
  • a one-term polynomial, such as 2x or 4x2, may also be called a "monomial" ("mono" meaning "one")
  • a two-term polynomial, such as 2x + y or x2 – 4, may also be called a "binomial" ("bi" meaning "two")
  • a three-term polynomial, such as 2x + y + z or x4 + 4x2 – 4, may also be called a "trinomial" ("tri" meaning "three")
I don't know if there are names for polynomials with a greater numbers of terms; I've never heard of any names other than what I've listed.
Not all polynomials have special names.

If you think you've got the hang of it try our Practice test:

Practice Test

1. Define the terms, using complete sentences (2 Marks):
a) Numeric co-efficient-
b) Constant-_
2. Complete the chart by filling in the blanks (9 Marks)
Polynomial
Degree
Numeric co-efficient (of first term)
Type of Polynomial
8



A^3-4b+14^2



6x^4 y^2-3x^3




3. Arrange -3x^4-7x^2+2+9x^3 in ascending powers of x (1 Mark).
_
b) Arrange 19x^3 y^4+20xy^7-5x^4 y^3 in descending powers of y (1 Mark).
_
4. Simplify each set of polynomials. Show your steps and work (4 Marks)
a) (3x-2x^2+9) + (-4x^2+5x-2) b) (8x^3-6x^2+8)-(-x^3+2x^2-3)





5. Expand and simplify. Show your steps and work (4 Marks).
a) 3(y-3) +2(y+8)-5(y+1) b) (-4d^2 e^3 f^5) ^3






6. Factor each polynomial. Leave answer in good form (6 Marks).
a) 9x+6y-3z b) 81ab-9bc+27bd
_
c) 30x^9 y^4 z^3+40x^5 y^6 z^5+60x^7 y^5 z^2

7. A cube has the following dimension. Find the volume by simplifying first! Then, use your simplified answer to find the volume if x=-2 (5 Marks).

-3x^3

Answers for the practice test:

Practice Test

1. Define the terms, using complete sentences (2 Marks):
a) Numeric co-efficient-a value (number) that when paired with a variable (the number comes before the letter)
b) Constant-A positive or negative number without a variable
2. Complete the chart by filling in the blanks (9 Marks)

Polynomial
Degree
Numeric co-efficient (of first term)
Type of Polynomial
8
0
8
Monomial
A^3-4b+14^2
3
1
Trinomial
6x^4 y^2-3x^3
6
6
Binomial

3. Arrange -3x^4-7x^2+2+9x^3 in ascending powers of x (1 Mark).
-7x^2 + 9x^3 -3x^4 +2
b) Arrange 19x^3 y^4+20xy^7-5x^4 y^3 in descending powers of y (1 Mark).
+20xy^7+19x^3 y^4-5x^4 y^3
4. Simplify each set of polynomials. Show your steps and work (2 Marks)
a) (3x-2x^2+9) + (-4x^2+5x-2)
=3x+5x-2x^2-4x^2+ (9-2)
=8x-6x^2+7
5. Expand and simplify. Show your steps and work (4 Marks).
a) 3(y-3) +2(y+8)-5(y+1) b) (-4d^2 e^3 f^5) ^3
=3y-9+27+16-5y-5 =-4^3 d^(2)(3) e^(3)(3) f(5)(3)
=3y+2y-5y-9+16-5 =-64d^6 e^9 f^15
=-1

6. Factor each polynomial. Leave answer in good form (6 Marks).
a) 9x+6y-3z b) 81ab-9bc+27bd
=3(3x+2y-z) =9b(9a-c+3d)

c) 30x^9 y^4 z^3+40x^5 y^6 z^5+60x^7 y^5 z^2
=10x^5 y^4 z^2 (3x^4 z+4y^2 z^3+6^2 y)
7. A cube has the following dimension. Find the volume by simplifying first! Then, use your simplified answer to find the volume if x=-2 (5 Marks).

(-3x^3)^3 -27x^9
=-3^3 x^(3)(3) =-27(-2)^9
=-27x^9 =-27(-512)
=13 824
Therefore the volume is 13 824 units if x=-2.


Use of Polynomials


  • to form polynomial equations, ranging from elementary problems to complicated science problems
  • used to define polynomial functions in chemistry, physics, economics, calculus, numerical analysis
  • to construct polynomial rings
  • to construct polynomial equation graphs
  • roller coaster polynomials
  • to find the perimeter of a figure

Beyond the Classroom

  • construction
  • financial planning
  • real estate
  • race car designs (National Hot Rod Association)
  • electronics
  • electricity
  • medicine



For more information please see Polynomials 2
Authors

Erica Z., Katie M., Ellen D., John M.

Resources**
Answer Corporation. (2008). What are Polynomials used for in a real world example? WikiAnswers. Retrieved Nov. 19, 2008, from http://wiki.answers.com/Q/What_are_polynomials_used_for_in_a_real_world_example

Pierce, R. (2008, Nov. 19). Polynomials. Math is Fun. Retrieved November 18, 2008, from http://www.mathsisfun.com/algebra/polynomials.html

Pierce, R. (2008, Nov. 19). Algerbra- Basic Definitions. Math is Fun. Retrieved December 18, 2008, from http://www.mathsisfun.com/algebra/definitons.html

Math Power 9 Ontario Edition. (1999). Toronto: McGraw-Hill Ryerson.