Factorization

F I N D I N G T H E PA RT S O F A MONOMIAL

FACTORIZATION &

GREATEST COMMON FACTORS

MONOMIALS 

• A Monomial is an algebraic expression containing only one term• Monomial - the prefix mono means one

• A monomial CANNOT contain a plus sign (+) or a minus (-) sign

• Two rules about monomials are:• A monomial multiplied by a monomial is also a monomial.• A monomial multiplied by a constant is also a monomial.

FACTORING WHOLE NUMBERS

FACTORS

• Factors are the numbers you multiply together to get a product.• The product 48 has many factors.

48 = 1 x 4848 = 2 x 2448 = 3 x 16

48 = 4 x 1248 = 6 x 8

FACTORING

• Factoring is the process of finding all the factors of a term.• It is like "splitting" an expression into a

multiplication of simpler expressions

6 3*2

10 2*5

20 2*10, 4*5

2555 5*511, 7*365, 35*73

HOW TO FIND THE FACTORS

To find all the factors • start at 1 and divide your number• if it can be divided write both 1 and the quotient

• move on to the number 2• again if it can be divided write 2 and the quotient

• If not divisible by 2 move on to 3• Continue this process until you reach a number you have

already written down• You can skip any numbers you are sure you can not divide

• 76/5 111/2 99/7

Remember we want only whole numbers

1*88, 2*44, 4*22, 8*11

FACTORING THE NUMBER 88

88/1 = 88 88/2 = 44 88/3 = X 88/4 = 22

88/5 = X 88/6 = X 88/7 = X 88/8 = 11

88/9 = X 88/10 = X 88/11 = repeated

number

We usually show our results like this:1, 2, 4, 8, 11, 22, 44, 88

FACTORING THE NUMBER 18

1 x 182 x 93 x 64 x ??5 x ??6 x 3Repeated numbers, we can stop.

The factors of 18 are: 1,2,3,6,9,18

Repeated numbers, we can stop.

If we cannot use 3, then we can skip the multiples of 3, and all the multiples of any number that we cannot use.

FACTORING THE NUMBER 250

1 x 2502 x 1253 x ??4 x ??5 x 50 The factors of 250

are:

1, 2, 5, 10, 25, 50, 125, 250

7 x ??10 x 2511 x 5013 x ??17 x ??19 x ??22 x ??23 x ??25 x 10

GREATEST COMMON FACTORWHOLE NUMBERS

USING GCF

• Some times to solve the problems in our lives we GCF without even knowing it

• Greatest Common Factor means• Greatest largest• Common same• Factor the numbers that are multiplied

together to make another number

HOW…

• Step one find all the factors18 24 66

18 1*18, 2*9, 3*61, 2, 3, 6, 9, 18

24 1*24, 2*12, 3*8, 4*61, 2, 3, 4, 6, 8, 12, 24

66 1*66, 2*33, 3*22, 4*14, 6*111, 2, 3, 4, 6, 11, 14, 22, 33, 66

HOW…

• Step 2 circle all the common factors

18 1, 2, 3, 6, 9, 18

24 1, 2, 3, 4, 6, 8, 12, 24

66 1, 2, 3, 4, 6, 11, 14, 22, 33, 66

Our common factors are: 1, 2, 3, 6The greatest common factor is: 6

Wait… there is another way…. A little bit easier too

PRIME NUMBERS

• Prime numbers are any number that can only be divided by 1 and itself. (except for the number 1)

• Here is a list of prime numbers

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101

• If a number is not prime we call it a composite number

PRIME FACTORING

Another way to do factoring is to use prime numbers

1. Divide the integer by 2, then continue doing so until you can not continue

2. Move to the next prime number and just like before keep dividing until you cannot, than move to the next prime number and repeat

3. You can stop when your Quotient is prime

Tip… As you go circle all the prime numbers

PRIME FACTORIZATION OF 100

100

2 X 502 is the first prime so we start with by dividing 100 by 2

Remember to circle the prime numbers. 50 can still be

divided by 2 so we continue.

2 X 25

25 is not divisible by 2. The next prime number (3) does not work either. We must continue trying prime numbers until one works.

Both numbers are prime, we can stop now.

5 X 5

EXAMPLES

10 18 25 2012 2555

/ \ / \ / \ / \ / \

2 5 2 9 5 5 2 1006 5 511

/ \ / \ / \

3 3 2 505 7 73

/ \

5 101

EXAMPLES

The Prime Factors are:

10 2 * 5

18 2 * 3 * 3

25 5 * 5

2020 2 * 2 * 5 * 101

2555 5 * 7 * 73

10 18 25 2020 2555

/ \ / \ / \ / \ / \

2 5 2 9 5 5 2 1010

5 511

/ \ / \ / \

3 3 2 505 7 73

/ \

5 101

GCF USING PRIME NUMBERS

• Step one find all the prime factors

18

2 9 2 12 2 33

3 3 2 6 3 11

2 3

24

66

GCF USING PRIME NUMBERS

• Step two find all the common prime factors

• 18 = 2 * 3 * 3

• 24 = 2 * 2 * 2 * 3

• 66 = 2 * 3 * 11

• The common prime factors are:

2 * 3 • and 2 x 3 = 6• As we saw before, the CGF

of 18, 24, and 66 is 6

ONE MORE EXAMPLE

• Step one find all the prime factors

56

2 28

2 14 2 42

2 14

2 7 2 21

28

84

2 7 3 7

ONE MORE EXAMPLE

• Let’s look at the prime factors

• 56 = 2 * 2 * 2 * 7

• 28 = 2 * 2 * 7

• 84 = 2 * 2 * 3 * 7

• The common prime factors are:

2 * 2 * 7 • and 2 x 2 x 7 = 28

• 56 / 28 = 2• 28 / 28 = 1• 84 / 28 = 3

FACTORING WITH VARIABLES

yy*y

y*y*yy*y*y*y

y*y*y*y*yy*y*y*y*y*y

y*y*y*y*y*y*yy*y*y*y*y*y*y*

y

FACTORING A VARIABLE

yy2

y3

y4

y5

y6

y7

y8

12345678

How many y’s? Factors

Notice a

pattern??

?

FACTORING MULTIPLE VARIABLES

Factors

xyxy2

x3y3

xyzx2y2y2

abc3de

x*yx*y*yx*x*x*y*y*yx*y*zx*x*y*y*z*za*b*c*c*c*d*e

It is j

ust that

easy

THE NEXT STEP

Prime factors of constant factors of variables

Final Results

xx

x*yx*y*y

x*x*y*yx*x*x*x*x*x*y*y*y*y*y*z*z*z

2132*3

2*2*32*2*2*2*2

2*2

2x13x6xy

12xy2

32x2y2

4x6y5z3

2*x13*x

2*3*x*y2*2*3*x*y*y

2*2*2*2*2*x*x*y*y2*2*x*x*x*x*x*x*y*y*y*y*y*z*z*z

GCF WITH CONSTANT & VARIABLES

• Step one find all the prime factors18x

2 9 2 12 2 33

3 3 2 6 3 11

2 3

24x3 66x2y x x3

x*x*x

x2y

x*x*y

GCF WITH CONSTANT & VARIABLES

• 18x2 * 2 * 3 * x

• 24 x3

2 * 2 * 2 * 3 * x * x * x• 66x2y

2 * 3 * 11 * x * x * y

The common prime factors are:2 * 3 * x

The greatest common factor is:6x

ONLINE RESOURCES

http://www.icoachmath.com

http://www.algebra-class.com

http://www.ixl.com/math/grade-8

http://examples.yourdictionary.com/examples-of-monomial.html

http://www.mathsisfun.com/algebra/factoring.html