Factoring



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Table of Contents (MARKED 2009/06/07)

 * 1) Factoring Video
 * 2) Factoring Powerpoint
 * 3) Common Terminology in Factoring
 * 4) What is a Factor
 * 5) Common Factors
 * 6) Greatest Common Factors
 * 7) Solving Quadritic Equations by using factoring
 * 8) Factoring Expression with Common Factors
 * 9) How does Factoring Relate to us in Real Life?

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Factoring Video
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Factoring Powerpoint


=Common Terminology In Factoring =

Here's some **terminology** that will come in handy when reading about or doing factoring.

**Algebraic Factoring**:Factoring is the process used to find the whole numbers by which a given number may be divided without remainder.

But there's also another type of factoring called "Invoice or Accounts Receivable Factoring" but we can forget about that for now, because we are focused on Algebraic Factoring.

**Common Factors**:Factors that are common to two or more numbers are said to be common factors.


 * Greatest Common Factors**:The greatest common factor, or GCF, is the greatest factor that divides two numbers or more numbers.


 * Prime Factorization**:Calculation of all prime factors in a number, usually used to find common factors and to help simplify fractions. Used when finding GCF(Greatest common factor).


 * Prime Factor**: A whole number greater than 1 that only has two whole number factors, 1 and itself.

= = =What is a factor? = =A factor is two or more numbers, that when multiplied together will give the given product.= =To factor a number is to find out what numbers have that number as a product.= There are 3 types of factors
 * Lowest Common Multiple**: The smallest multiple that is exactly divisible by every number in a set of numbers.
 * Common Factor
 * Greatest Common Factor
 * Prime Factor


 * Use tree diagrams to find the factors of a number*
 * A number can have multiple factors*
 * When you multiply the factors together they should equal the original number

Example #1 The factors of 10 are 5 and 2. You can check to see if your answer is right by multipling 2 by 5 which equals 10 meaning you did it right. 10 / \ 5 2 Example #2 The factors of 6 are 3 and 2 and you can reduce them even further. 6 / \ 3 2 Example #3: The factors for 16 could be 4 and 4, but if you were to reduce it even more it would be 2 x 2 x 2 x 2 or 2^4 16 / \ 4 4 / \ / \ 2 2 2 2

=Common Factor = Common factors is a common number between the two numbers your factoring. To find these common factors you first list all the factors for your numbers. Next you find all the factors that are common with your numbers and list them. Prime numbers can not have a common factor except for 1 and themselves, for there is no other factored numbers that go in to the prime number.
 * Prime numbers and common factors**



Example 8 Find the common factors of 26 and 39.

Solution:
So, the common factors of 26 and 39 are 1 and 13

=Greatest Common Factor =

The greatest common factor, or GCF, of two integers is the largest positive integer that devides both numbers without a remainder. To find the GCF of two numbers:

The simplest way to show you how to get the GCF is the following: Step 1. List the prime factors of each number Step 2. See the all the factors they have in common and multiply the common factors together, and if the numbers have no common factors then the greatest common factor is 1. when determining the GCF of a number you must first write each number as a product of its prime factors. Example: == =__Prime Factors __= =The "Prime Factors" of a number are the prime numbers that devide into an iteger exsactly, without leaving a remainder.= = = = This comes in handy, when solving problems; to find the largest factor dividing 2 or more numbers. =

Example and how to work with prime factors ..




 * EXAMPLE #1.**

Here's one way to find the GCF. And that is by, taking the number, and writing down all the numbers that make up the number, and just look for the number they both have in common.

25- 1, 5, and 25 75-1, 3, 5, 15, 25, and 75.

In this case, there is 5 that is the highest number they both have in common, making it the GCF.


 * EXAMPLE #2.**

Let me explain this to you in more depth, so lets do a problem together. Let's find the factors of 30 and 75. Let's break them down into prime factors now though. For this we can use the tree diagram way. 75 / \ 3 25 / / \ 3 5 5 they equal the number you started with you have the right number, but make sure the numbers are PRIME and can't be broken down anymore!
 * Can't be broken down any further, meaning they are now prime numbers.
 * To check if this answer is correct multiply all the numbers together, and if

Checking my answer:


 * 2 can't be broken down anymore
 * 5 can't be broken down anymore
 * 3 can't be broken down anymore
 * 2x5x3=30 meaning we have the right numbers!

75 30 / \ / \ 3 25 2 15 / / \ / / \ 3 5 5 2 <span style="background-color: rgb(227, 235, 0);">3 <span style="color: rgb(250, 0, 0); background-color: rgb(227, 235, 0);"> <span style="background-color: rgb(227, 235, 0);">5

Step 1. Highlight the prime numbers they have in common. Step 2. Multiply the prime factors that they both have in common

3x5=15

Step 3. The number you get once you have multiplied the prime factors that make up the 2 or more numbers, is the Greatest Common Factor (GCF). Step 4. Give yourself a pat on the back, you've the answer! Step 5. Move on to the next question, and just follow the process above, even if its with more than 2 numbers.

To "factor" is to express a number as the product of two or more numbers, or an algebraic expression as the product of two or more algebraic expressions.
 * __Common Factors and the GCF__ **

The "Prime Factors" of a number are the prime numbers that divide into an integer exactly, without leaving a remainder. The "Greatest Common Factor", or GCF, of two non-zero integers is the largest positive integer that divides the both of the numbers without a remainder.

The first four prime numbers are: 2, 3, 5, and 7.

When determining the Greatest Common Factor of a number, you must first write each number as a product of it's prime factors. The <span style="background-color: rgb(255, 255, 0);"> greatest common factor of two (or more) integers is the largest integer that is a factor of both (or all) numbers. **<span style="color: rgb(0, 0, 0);">Consider the numbers 18, 24, and 36. **<span style="color: rgb(255, 0, 0);"> The greatest common factor is 6. <span style="font-size: 90%; color: rgb(0, 0, 0);">(6 is the largest integer that will divide evenly into all three numbers)

The greatest common factor of two (or more) monomials is the product of the greatest common factor of the numerical coefficients (the numbers out in front) and the highest power of every variable that is a factor of each monomial. **<span style="color: rgb(0, 0, 0);">Consider: 10x2y3 and 15xy2 **<span style="color: rgb(255, 0, 0);"> The greatest common factor is 5xy2. (the largest factor of 10 and 15 is 5, the highest power of x that is contained in both terms is x, and the highest power of y that is contained in both terms is y2 ) <span style="color: rgb(0, 0, 0);">

When factoring polynomials, you must look for the largest monomial which is a factor of each of the polynomials. **Factor: 4x + 8y** The largest integer that will divide evenly into 4 and 8 is 4. Since the terms do not contain a variable (x or y) in common, we cannot factor any variables. <span style="color: rgb(255, 0, 0);">**Answer: 4(x + 2y)**



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=__<span style="background-color: rgb(0, 255, 240);">Here is an example of how to solve quadratic equations by using factoring __ .... = =<span style="font-family: Tahoma,Geneva,sans-serif; background-color: rgb(67, 208, 199);"> = =<span style="font-family: Tahoma,Geneva,sans-serif; background-color: rgb(67, 208, 199);">Factoring Expressions with Common Factors = <span style="font-family: 'Times New Roman',Times,serif;"> = = <span style="font-family: 'Times New Roman',Times,serif;"> To factor an expression with common factors, follow this example:

Factor the expression: 6//x²// - 14//x//. First, determine the Greatest Common Factor of both terms. Then, divide both terms by the GCF. 6//x²// = **<span style="background-color: rgb(240, 255, 0);">2 ** x **<span style="background-color: rgb(221, 220, 14);">3 ** x **//<span style="background-color: rgb(243, 233, 18);">x //** x //**<span style="background-color: rgb(244, 237, 21);">x **// 14//x// = **2** x **7** x **//x//** The GCF is **2//x//**.

The second factor is __6//x²//__ _ __14//x//__ or 3//x// - 7. 2//x// 2//x// The factors of 6//x²// - 14//x// are 2//x// and 3//x// - 7. Therefore, 6//x²// - 14//x// = 2//x//(3//x// - 7).

You should be sure to double-check your answer and make sure it's correct. You can do that by expanding.

2//x//(3//x// - 7) = 2//x//(3//x// - 7) = 6//x²// - 14//x// <span style="color: rgb(130, 32, 238);">* To check your answer, use the answer you found when finding the factors of your expression, and simply apply the Distributive Property (a.k.a. the Rainbow Rule) to those factors and you should end up with the expression you started out with. In this case, it was //6x//² - //14x// ... which was what the expansion came out to be! Wasn't that easy?

Example #1 example of factoring out X 3x^3+2x^2+x x(3x^2+2+1)

Example #2 25y 5x5xy hcf = 5

Examples on how to factor polynomials. example #1 3x^2 z – 3zx – 18z = 3z(x² - x - 6)

= 3z(x + 2)(x - 3)

example #2 x^4 – y^4 x^4 - y^4

= (x² - y²)(x² + y²)

= (x - y)(x + y)(x - yi)(x + yi)

=<span style="font-family: Tahoma,Geneva,sans-serif; background-color: rgb(67, 208, 199);">How does Factoring Relate to us in Real Life? =

Try this, if your having a party for 3 kids, and you have 1 pizza(8 slices) and 3 hot dogs. You have to figure out how to split it into everyone evenly. Its like following a recipe and adjusting it so everyone has an equal amount. Or if you catered for a certain amount of people and you have to figure out how much food each person could have if you want to plate it a certain way, and not have it a buffet style.

Also another example that I'll put in question form is If Joe attends the gym every third day, Mark attends every second day and Bill attends every fifth day, when will they next be at the gym on the same day if this last occurred 7 days ago ?

You can do 2 things, either do the table done below, or just simply multiply 2x3x5 because they are all prime numbers and you be able to figure out their lowest common multiple, which is 30 and then the factors of 30 are 2 5 and 7 which is how you can check your answer.


 * Monday || Tuesday || Wednesday || Thursday || Friday || Saturday || Sunday ||
 * <span style="background-color: rgb(211, 18, 18);">Joe Mark Bill ||  || Mark || Joe || Mark || Bill || Joe Mark ||
 * || Mark || Joe || Mark Bill ||  || Joe Mark ||   ||
 * Mark || Joe Bill || Mark ||  || Joe Mark ||   || Mark Bill ||
 * Joe || Mark ||  || Joe Mark || Bill || Mark || Joe ||
 * Mark ||  || <span style="background-color: rgb(211, 18, 18);">Joe Mark Bill ||   || Mark || Joe || Mark ||
 * Bill || Joe Mark ||  || Mark || Joe || Mark Bill ||   ||

So the next time all three of them will attend the gym at the same time is in 30 days of their first little get together. But since it last occurred 7 days ago, you have to subtract the 7 days from the 30 days.

Another real life example would be, of course HOW COULD WE FORGET!? Our math teachers use factors in their lessons, well we know for sure when they teach factoring lessons and also when they teach some other subjects like polynomials and such.

But these are only a few examples, there are MANY, MANY more that these, but these are a few examples of when you would use factoring in real life.

=<span style="font-size: 120%; color: rgb(245, 5, 5); font-family: 'Times New Roman',Times,serif;">**Citation** = =<span style="font-family: Tahoma,Geneva,sans-serif; background-color: rgb(67, 208, 199);">Knill, G. (1999). //Math Power 9//. Toronto, Ontario: McGraw-Hill Ryerson Limited. = <span style="font-family: Tahoma,Geneva,sans-serif;"> =<span style="font-family: Tahoma,Geneva,sans-serif; background-color: rgb(67, 208, 199);"> = =<span style="font-family: Tahoma,Geneva,sans-serif; background-color: rgb(67, 208, 199);">//Greatest Common Factors (GCF)// (1999, May 9). Retrieved December 8, 2008, from http://www.math.com/school/subject1/lessons/S1U3L2DP.html = mathwarehouse,com. (n.d.). mathwarehouse.com. [] =<span style="font-family: Tahoma,Geneva,sans-serif; background-color: rgb(67, 208, 199);">//Common Factoring// (2003, May). Retrieved December 7, 2008, from http://www.curriculum.org/tcf/teachers/projects/algebra/commonfactorflash.adp =

=<span style="font-family: Tahoma,Geneva,sans-serif; background-color: rgb(67, 208, 199);">//Algebra Factoring// (1999, May 12). Retrieved December 10, 2008, from http://library.thinkquest.org/20991/alg/factoring.html = =<span style="font-family: Tahoma,Geneva,sans-serif; background-color: rgb(67, 208, 199);">Malik, M. (2008). //Math Notes//. Hamilton: Momna Malik. = =<span style="font-family: Tahoma,Geneva,sans-serif; background-color: rgb(67, 208, 199);"> = math.com. (n.d.). Author. =**__<span style="font-size: 130%; color: rgb(209, 5, 5); font-family: 'Times New Roman',Times,serif;">Group: __ Momna Malik, Robyn Wallace, Carly Butter, Zack O'Leary**= =__<span style="color: rgb(0, 0, 0);">**<span style="color: rgb(211, 18, 18);">Group #2 :** __<span style="color: rgb(0, 0, 0);"> **James P., Michelle Smith, Josh Malka, Derek Gabbani** <span style="color: rgb(211, 18, 18);"> = =<span style="font-family: Tahoma,Geneva,sans-serif; background-color: rgb(67, 208, 199);"> =
 * //Factoring// (n.d.). Retrieved December 14, 2008, from [|http://home.avvanta.com/~math/def2.cgi?t=factor****]**