Polynomials+1

=__Polynomials__ =
 * __ By: Ayman M. Chris G. Sara G. Marisa U. __**
 * 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.

Here is some terminology used when working with polynomials:
 * __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 ||


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

1) Numeric Co-effcient (a number) __5__x^2 2) Literal Co-effcient (a letter)5__x__^2
 * Term **: A term is made up of two parts * Terms are Separated by + or a - sign.


 * Algebraic Expression: ** expressions that include numbers and variables

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
 * Like Terms: ** Terms which have the same variable and power. Like terms can be combined using addition or subtraction.


 * 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 3//x//4 or 6//x//. 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: a polynomial term... || ...because the variable has a negative exponent. || a polynomial term... || ...because the variable is in the denominator. || a polynomial term... || ...because the variable is inside a radical. ||
 * 6//x// –2 || This is NOT
 * 1///x//2 || This is NOT
 * <span style="color: rgb(0,0,0); font-family: Times New Roman;">//sqrt//(//x//) || <span style="color: rgb(0,0,0); font-family: Arial;">This is NOT
 * <span style="color: rgb(0,0,0); font-family: Times New Roman;">4//x//2 || <span style="color: rgb(0,0,0); font-family: Arial;">This IS a polynomial term... || <span style="color: rgb(0,0,0); font-family: Arial;">...because it obeys all the rules. ||

<span style="color: rgb(0,0,0); font-family: Arial;">Here is a typical polynomial: <span style="color: rgb(0,0,0);"> 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 <span style="color: rgb(255,0,0);">"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 <span style="color: rgb(255,0,0);">"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: <span style="color: rgb(0,0,0); font-family: Arial;">This polynomial has four terms, including a fifth-degree term, a third-degree term, a first-degree term, and a constant term. <span style="color: rgb(128,0,128); font-family: Arial;">**This is a fifth-degree polynomial.** <span style="color: rgb(0,0,0); font-family: Arial;">This polynomial has three terms, including a fourth-degree term, a second-degree term, and a first-degree term. There is no constant term. <span style="color: rgb(128,0,128); font-family: Arial;">**This is a fourth-degree polynomial.**
 * <span style="color: rgb(0,128,0);">** Give the degree of the following polynomial: 2//x//5 – 5//x//3 – 10//x// + 9 **
 * **<span style="color: rgb(0,128,0);"> Give the degree of the following polynomial: 7//x//4 + 6//x//2 + //x// **

<span style="font-size: 10pt; color: rgb(77,13,242); font-family: Verdana;"> Monomial, binomial and trinomial are specials names given to polynomials with a certain number of terms. To remember...Think Monocycle! <span style="font-size: 10pt; color: rgb(44,187,17); font-family: Verdana;">A monomial: 2x <span style="font-size: 10pt; color: rgb(237,10,7); font-family: Verdana;">Not a monomial: 2x-1 || An expression containing __two__ terms is called a ** binomial ** To remember...Think Bicycle! <span style="font-size: 10pt; color: rgb(44,187,17); font-family: Verdana;">A binomial: 8x-3 <span style="font-size: 10pt; color: rgb(237,10,7); font-family: Verdana;">Not a binomial: 8x-3+10 || An expression containing __three__ terms is called a ** trinomial **. To remember...Think Tricycle! <span style="font-size: 10pt; color: rgb(44,187,17); font-family: Verdana;">A trinomial: 10x-5+4 <span style="font-size: 10pt; color: rgb(237,10,7); font-family: Verdana;">Not a trinomial: 10x-5 || We name polynomials based on the number of terms. 2x <span style="font-size: 10pt; color: rgb(255,0,0); font-family: Verdana;">One Term Monomial 8x-3 <span style="font-size: 10pt; color: rgb(255,0,0); font-family: Verdana;">Two Terms Binomial 10x-5+4 <span style="font-size: 10pt; color: rgb(255,0,0); font-family: Verdana;">Three Terms Trinomial
 * <span style="font-size: 10pt; color: rgb(6,4,4); font-family: Verdana;">__Monomials, Binomials, Trinomials__ **<span style="font-size: 10pt; color: rgb(6,4,4); font-family: Verdana;">
 * =====**<span style="font-size: 11pt; color: rgb(28,11,168); font-family: Verdana;">Monomials ** ===== || **<span style="font-size: 11pt; color: rgb(28,11,168); font-family: Verdana;">Binomials ** || **<span style="font-size: 11pt; color: rgb(28,11,168); font-family: Verdana;">Trinomials ** ||
 * An expression containing __one__ term is called a ** monomial **.
 * Polynomial Names**

<span style="font-size: 10pt; color: rgb(11,11,224); font-family: Verdana;">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 <span style="font-size: 10pt; color: rgb(255,3,0); font-family: Verdana;">/ 3. Since there are two parts to this equation that means that it is a binomial.

**<span style="color: rgb(177,37,110); font-family: Verdana;">__Polynomial__ **
Any polynomial with 4 or more terms is called a polynomial. <span style="font-size: 10pt; color: rgb(195,24,105); font-family: Verdana;"> __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.**

<span style="color: rgb(241,55,218); font-family: Verdana;">__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__ <span style="font-size: 10pt; color: rgb(255,0,102); font-family: Verdana;"> 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.
 * <span style="font-size: 10pt; color: rgb(30,255,0); font-family: Verdana;">__Ascending Order__ <span style="font-size: 10pt; color: rgb(255,0,102); font-family: Verdana;">

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 <span style="font-size: 10pt; color: rgb(102,0,255); font-family: Verdana;">. <span style="color: rgb(11,11,224);">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 * <span style="font-size: 10pt; color: rgb(5,5,5); font-family: Verdana;">4^8, 3^2, 7^5, 2^4

<span style="color: rgb(255,102,51); font-family: Verdana;">__Solving:__
<span style="font-size: 10pt; color: rgb(255,102,51); font-family: Verdana;"> 3^2 <span style="font-size: 10pt; color: rgb(255,0,51); font-family: Verdana;">, <span style="font-size: 10pt; color: rgb(255,102,51); font-family: Verdana;"> 2^4, 7^5, 4^8
 * **4^8 = 4x4x4x4x4x4x4x4=65536**
 * **7^5 = 7x7x7x7x7=16807**
 * **2^4 = 2x2x2x2=16**
 * **3^2 = 3x3=9**
 * The numbers arranged in Ascending Order are**
 * <span style="font-size: 10pt; color: rgb(255,102,51); font-family: Verdana;">

<span style="font-size: 10pt; color: rgb(255,0,51); font-family: Verdana;">__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. <span style="font-size: 10pt; color: rgb(255,0,51); font-family: Verdana;"> ** <span style="font-size: 10pt; color: rgb(255,0,163); font-family: Verdana;">__Descending Order__ <span style="font-size: 10pt; color: rgb(51,204,102); font-family: Verdana;"> <span style="font-size: 10pt; color: rgb(255,0,51); font-family: Verdana;"> __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. <span style="font-size: 10pt; color: rgb(51,0,255); font-family: Verdana;"> <span style="font-size: 10pt; color: rgb(6,5,5); font-family: Verdana;">E.g. 12, 8, 5, 2, and 1 are arranged in Descending order. 12 being the biggest, and 1 being the smallest. <span style="font-size: 10pt; color: rgb(51,0,255); font-family: Verdana;"> <span style="font-size: 10pt; color: rgb(6,5,5); font-family: Verdana;">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 <span style="font-size: 10pt; color: rgb(51,0,255); font-family: Verdana;">. <span style="font-size: 10pt; color: rgb(255,0,163); font-family: Verdana;"> This is an Example 3^3, 2^5, 2^4 <span style="font-size: 10pt; color: rgb(255,102,51); font-family: Verdana;"> ** <span style="font-size: 10pt; color: rgb(255,102,204); font-family: Verdana;">__Solving__

The numbers arranged in Descending Order are: 2^5, 3^3, 2^4
 * <span style="font-size: 10pt; color: rgb(255,102,204); font-family: Verdana;">**3^3 =3x3x3=27**
 * <span style="font-size: 10pt; color: rgb(255,102,204); font-family: Verdana;">**2^5 =2x2x2x2x2=32**
 * <span style="font-size: 10pt; color: rgb(252,85,223); font-family: Verdana;">**2^4 =2x2x2x2=16**
 * <span style="font-size: 10pt; color: rgb(255,102,204); font-family: Verdana;">

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

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: 7** x ** + ** x ** = 8** x ** 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__

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

8x^2 y^5x^3 y^2zx^6 y^2zx^6 y^5x^3
 * Then put them in** ascending order **: (REMEMBER: ASCENDING MEANS SMALLEST TO BIGGEST)**
 * Then, put them in** descending order **(REMEMBER: DESCENDING MEANS BIGGEST TO SMALLEST)**
 * 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 <span style="color: rgb(221,54,172); font-family: Verdana;">

and all Z terms to the other side of the equal sign <span style="color: rgb(221,54,172); font-family: Verdana;">

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.

<span style="font-size: 10pt; color: rgb(219,81,116); font-family: Verdana;"> Unlike Terms<span style="font-size: 10pt; color: rgb(219,81,116); font-family: Verdana;">

Unlike terms are really easy

They are the exact opposite of like terms

<span style="color: rgb(220,50,90); font-family: Verdana;">__8x, 8z-__ these are unlike terms because the have a different variable

__<span style="color: rgb(220,50,90); font-family: Verdana;">9 <span style="color: rgb(220,50,90); font-family: 'MS Mincho';">⁶ <span style="color: rgb(220,50,90); font-family: Verdana;">, 9 <span style="color: rgb(220,50,90); font-family: 'MS Mincho';">⁴ <span style="color: rgb(220,50,90); font-family: Verdana;"> - __<span style="color: rgb(220,50,90); font-family: Verdana;"> these are unlike terms because they are raised to different exponents

<span style="font-size: 13pt; color: rgb(215,20,121); font-family: Verdana;"> <span style="color: rgb(0,0,0); font-family: Arial;">When a term contains both a number and a variable part, the number part is called the <span style="color: rgb(255,0,0);">"coefficient". The coefficient on the leading term is called the <span style="color: rgb(255,0,0);">"leading" coefficient. <span style="color: rgb(0,0,0);"> 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. <span style="color: rgb(255,255,255); font-family: Arial;"> Copyright © Elizabeth Stape006-2008 All Rights Reserved <span style="color: rgb(0,0,0); font-family: Arial;">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: <span style="color: rgb(0,0,0); font-family: Arial;">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: 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)
 * <span style="color: rgb(0,0,0);"> a one-term polynomial, such as 2//x// or 4//x//2, may also be called a "monomial" ("mono" meaning "one")
 * <span style="color: rgb(0,0,0);"> a two-term polynomial, such as 2//x// + //y// or //x//2 – 4, may also be called a "binomial" ("bi" meaning "two")
 * <span style="color: rgb(0,0,0);"> a three-term polynomial, such as 2//x// + //y// + //z// or //x//4 + 4//x//2 – 4, may also be called a "trinomial" ("tri" meaning "three")
 * __ Practice Test __**
 * 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) 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
 * __
 * 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 ||

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

__Beyond the Classroom__
 * 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


 * 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

<span style="font-family: Verdana,Geneva,sans-serif;">//Math Power 9 Ontario Edition//. (1999). Toronto: McGraw-Hill Ryerson.