The document discusses factoring monomials and finding the greatest common factor of whole numbers and expressions with variables. It defines monomials as expressions with one term and no operators. It explains that factoring is splitting an expression into a multiplication of simpler expressions by finding all factors. The greatest common factor is the largest common factor among expressions. It provides examples of factoring whole numbers and expressions with variables by finding prime factors and common factors.

1. FACTORIZATION
&
GREATEST COMMON FACTORS
FINDING THE PARTS OF A MONOMIAL

2. 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.

3. FACTORING
WH O L E N U M B E R S

4. FACTORS
• Factors are the numbers you multiply together to
get a product.
• The product 48 has many factors.
48 = 1 x 48 48 = 4 x 12
48 = 2 x 24 48 = 6 x 8
48 = 3 x 16

5. 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

6. 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

7. 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
1*88, 2*44, 4*22, 8*11
We usually show our results like this:
1, 2, 4, 8, 11, 22, 44, 88

8. FACTORING THE NUMBER 18
1 x 18 The factors of 18
2x9 are: 1,2,3,6,9,18
3x6
4 x ??
5 x ??
6x3
Repeated numbers, we can
stop.

9. FACTORING THE NUMBER 250
1 x 250
2 x 125
3 x ?? If we cannot use 3, then we
4 x ?? can skip the multiples of 3,
The factors of 250
5 x 50 and all the multiples of any
7 x ?? number that we cannot use. are:
10 x 25
11 x 50 1, 2, 5, 10, 25,
13 x ??
17 x ??
50, 125, 250
19 x ??
22 x ??
23 x ??
25 x 10 Repeated numbers, we can
stop.

10. GREATEST COMMON
FACTOR
WH O L E N U M B E R S

11. 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

12. HOW…
• Step one find all the factors
18 24 66
18 1*18, 2*9, 3*6
1, 2, 3, 6, 9, 18
24 1*24, 2*12, 3*8, 4*6
1, 2, 3, 4, 6, 8, 12, 24
66 1*66, 2*33, 3*22, 4*14, 6*11
1, 2, 3, 4, 6, 11, 14, 22, 33, 66

13. 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, 6
The greatest common factor is: 6
Wait… there is another way…. A little bit easier too

14. 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
• If a number is not prime we call it a composite

15. 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

16. PRIME FACTORIZATION OF 100
100
2 is the first prime so
2 X 50
we start with by
Remember to
dividing 100 by 2
circle the prime
numbers. 50 can still be divided
2 continue.
by 2 so we
X 25
25 is not divisible by 2. The next
5 X 5
prime number (3) does not work
either. We must continue trying
Both numbers are
prime numberscan stop works.
prime, we until one
now.

17. EXAMPLES
10 18 25 2012 2555
/ / / / /
2 5 2 9 5 5 2 1006 5 511
/ / /
3 3 2 505 7 73
/
5 101

18. EXAMPLES
10 18 25 2020 2555 The Prime Factors are:
/ / / / / 10 2*5
2 5 2 9 5 5 2 1010 5 511
18 2*3*3
/ / /
25 5*5
3 3 2 505 7 73
2020 2 * 2 * 5 * 101
/
5 101 2555 5 * 7 * 73

19. GCF USING PRIME NUMBERS
• Step one find all the prime factors
18 24 66
2 9 2 12 2 33
3 3 2 6 3 11
2 3

20. GCF USING PRIME NUMBERS
• Step two find all the common prime factors
• 18 = 2 * 3 * 3 • The common prime factors are:
• 24 = 2 * 2 * 2 * 3 2*3
• 66 = 2 * 3 * 11 • and 2 x 3 = 6
• As we saw before, the CGF
of 18, 24, and 66 is 6

21. ONE MORE EXAMPLE
• Step one find all the prime factors
56 28 84
2 28 2 14 2 42
2 14 2 7 2 21
2 7 3 7

22. ONE MORE EXAMPLE
• Let’s look at the prime factors
• 56 = 2 * 2 * 2 * 7 • The common prime factors are:
• 28 = 2 * 2 * 7 2*2*7
• 84 = 2 * 2 * 3 * 7 • and 2 x 2 x 7 = 28
• 56 / 28 = 2
• 28 / 28 = 1
• 84 / 28 = 3

23. FACTORING
WI T H V A R I A B L E S

24. FACTORING A VARIABLE
How many y’s? Factors
y 1 y
y2 2 y*y
y3 3 y*y*y
y4 4 y*y*y*y
y5 5 y*y*y*y*y
y6 6 y*y*y*y*y*y
y7 7 y*y*y*y*y*y*y
y8 8 y*y*y*y*y*y*y*y

25. FACTORING MULTIPLE VARIABLES
Factors
xy x*y
xy2 x*y*y
x3y3 x*x*x*y*y*y
xyz x*y*z
x2y2y2 x*x*y*y*z*z
abc3de a*b*c*c*c*d*e

26. THE NEXT STEP
Prime factors of
factors of variables
constant
2x 2 x
13x 13 x
6xy 2*3 x*y
12xy2 2*2*3 x*y*y
32x2y2 2*2*2*2*2 x*x*y*y
4x6y5z3 2*2 x*x*x*x*x*x*y*y*y*y*y*z*z*z
Final Results
2*x
13*x
2*3*x*y
2*2*3*x*y*y
2*2*2*2*2*x*x*y*y
2*2*x*x*x*x*x*x*y*y*y*y*y*z*z*z

27. GCF WITH CONSTANT & VARIABLES
• Step one find all the prime factors
18x
x 24x3
x 66x2y
x
2 9 2 12 2 33
3 3 2 6 3 11
x*x*y
2 3
x*x*x

28. GCF WITH CONSTANT & VARIABLES
• 18x
2*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

29. 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