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.

Lowest Common Multiple: The smallest multiple that is exactly divisible by every number in a set of numbers.

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

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 and common factors
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.

The common factors of 4 and 6 are 1 and 2. This is as the factors of 4 are 1, 2 and 4 and the factors of 6 are 1, 2, 3 and 6.

Example 8
Find the common factors of 26 and 39.

Solution:

The common factors are in bold.

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
*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
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!

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 3 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.

Common Factors and the GCF 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.

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 greatest common factor of two (or more) integers is the largest integer that is a factor of both (or all) numbers. Consider the numbers 18, 24, and 36.
The greatest common factor is 6. (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. Consider: 10x2y3 and 15xy2
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 )

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. Answer: 4(x + 2y)

Factoring the Time

Here is an example of how to solve quadratic equations by using factoring ....

Factoring Expressions with Common Factors

To factor an expression with common factors, follow this example:

Factor the expression: 6x² - 14x.
First, determine the Greatest Common Factor of both terms.
Then, divide both terms by the GCF.
6x² = 2 x 3 x x x x
14x = 2 x 7 x x
The GCF is 2x.

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

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

2x(3x - 7)
= 2x(3x - 7)
= 6x² - 14x * 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 6x² - 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)

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

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

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.

Citation

Knill, G. (1999). Math Power 9. Toronto, Ontario: McGraw-Hill Ryerson Limited.

Factoring## Table of Contents (MARKED 2009/06/07)

## Factoring Video

## Factoring Powerpoint

## Common Terminology In Factoring

Here's some

terminologythat 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.Lowest Common Multiple: The smallest multiple that is exactly divisible by every number in a set of numbers.## 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 factorsExample #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 and common factorsPrime 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.

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

*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

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!

Checking my answer:

75 30

/ \ / \

3 25 2 15

/ / \ / / \

3 5 5 2 3 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.

Common Factors and the GCFTo "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.

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 greatest common factor of two (or more) integers is the largest integer that is a factor of both (or all) numbers.

Consider the numbers 18, 24, and 36.The greatest common factor is 6.

(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.

Consider: 10x2y3 and 15xy2The 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 )

When factoring polynomials, you must look for the largest monomial which is a factor of each of the polynomials.

Factor: 4x + 8yThe 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.

Answer: 4(x + 2y)Here is an example of how to solve quadratic equations by using factoring....## Factoring Expressions with Common Factors

Factor the expression: 6

x²- 14x.First, determine the Greatest Common Factor of both terms.

Then, divide both terms by the GCF.

6

x²=2x3xxxx14

x=2x7xxThe GCF is

2.xThe second factor is

6_x²14or 3xx- 7.2

x2xThe factors of 6

x²- 14xare 2xand 3x- 7.Therefore, 6

x²- 14x= 2x(3x- 7).You should be sure to double-check your answer and make sure it's correct. You can do that by expanding.

2

x(3x- 7)= 2

x(3x- 7)= 6

x²- 14x* 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

6x² -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)

## 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.

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.

Citation## Knill, G. (1999).

Math Power 9. Toronto, Ontario: McGraw-Hill Ryerson Limited.

mathwarehouse,com. (n.d.). mathwarehouse.com.Greatest Common Factors (GCF)(1999, May 9). Retrieved December 8, 2008, from http://www.math.com/school/subject1/lessons/S1U3L2DP.htmlhttp://imgs.xkcd.com/comics/factoring_the_time.png

Common Factoring(2003, May). Retrieved December 7, 2008, from http://www.curriculum.org/tcf/teachers/projects/algebra/commonfactorflash.adpAlgebra Factoring(1999, May 12). Retrieved December 10, 2008, from http://library.thinkquest.org/20991/alg/factoring.html## Malik, M. (2008).

Math Notes. Hamilton: Momna Malik.Factoring(n.d.). Retrieved December 14, 2008, from http://home.avvanta.com/~math/def2.cgi?t=factor****Group:Momna Malik, Robyn Wallace, Carly Butter, Zack O'LearyGroup #2:James P., Michelle Smith, Josh Malka, Derek Gabbani