Factoring



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?


Factoring Video

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.

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.
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.
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:

factor-GCF.gif

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 ..

external image S1U3L1ex.gif


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.

external image bug.gifConsider 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.
external image bug3.gifConsider: 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.
external image bug2.gifFactor: 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
Factoring the Time


external image moz-screenshot.jpgexternal image moz-screenshot-1.jpg


Here is an example of how to solve quadratic equations by using factoring .... external image picture-steps-solve-factor.gif

Factoring Expressions with Common Factors


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

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

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


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.
http://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.adp


Algebra 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.

math.com. (n.d.). Author.
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'Leary

Group #2: James P., Michelle Smith, Josh Malka, Derek Gabbani

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