Factorization
Transcript of Factorization
![Page 1: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/1.jpg)
F I N D I N G T H E PA RT S O F A MONOMIAL
FACTORIZATION &
GREATEST COMMON FACTORS
![Page 2: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/2.jpg)
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.
![Page 3: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/3.jpg)
FACTORING WHOLE NUMBERS
![Page 4: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/4.jpg)
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
![Page 5: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/5.jpg)
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
![Page 6: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/6.jpg)
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
![Page 7: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/7.jpg)
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
![Page 8: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/8.jpg)
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
![Page 9: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/9.jpg)
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
![Page 10: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/10.jpg)
GREATEST COMMON FACTORWHOLE NUMBERS
![Page 11: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/11.jpg)
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
![Page 12: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/12.jpg)
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
![Page 13: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/13.jpg)
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
![Page 14: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/14.jpg)
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
![Page 15: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/15.jpg)
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
![Page 16: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/16.jpg)
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
![Page 17: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/17.jpg)
EXAMPLES
10 18 25 2012 2555
/ \ / \ / \ / \ / \
2 5 2 9 5 5 2 1006 5 511
/ \ / \ / \
3 3 2 505 7 73
/ \
5 101
![Page 18: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/18.jpg)
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
![Page 19: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/19.jpg)
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
![Page 20: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/20.jpg)
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
![Page 21: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/21.jpg)
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
![Page 22: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/22.jpg)
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
![Page 23: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/23.jpg)
FACTORING WITH VARIABLES
![Page 24: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/24.jpg)
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??
?
![Page 25: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/25.jpg)
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
![Page 26: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/26.jpg)
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
![Page 27: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/27.jpg)
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
![Page 28: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/28.jpg)
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
![Page 29: Factorization](https://reader036.fdocuments.in/reader036/viewer/2022081519/558c2a6cd8b42a9b568b45d6/html5/thumbnails/29.jpg)
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