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Factorization

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Factorization

1. 1. FACTORIZATION &GREATEST COMMON FACTORSFINDING THE PARTS OF A MONOMIAL
2. 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. 3. FACTORINGWH O L E N U M B E R S
4. 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. 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. 6. HOW TO FIND THE FACTORSTo 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. 7. FACTORING THE NUMBER 8888/1 = 88 88/2 = 44 88/3 = X 88/4 = 2288/5 = X 88/6 = X 88/7 = X 88/8 = 1188/9 = 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. 8. FACTORING THE NUMBER 181 x 18 The factors of 182x9 are: 1,2,3,6,9,183x64 x ??5 x ??6x3Repeated numbers, we canstop.
9. 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 2511 x 50 1, 2, 5, 10, 25,13 x ??17 x ?? 50, 125, 25019 x ??22 x ??23 x ??25 x 10 Repeated numbers, we can stop.
10. 10. GREATEST COMMON FACTOR WH O L E N U M B E R S
11. 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. 12. HOW…• Step one find all the factors 18 24 6618 1*18, 2*9, 3*6 1, 2, 3, 6, 9, 1824 1*24, 2*12, 3*8, 4*6 1, 2, 3, 4, 6, 8, 12, 2466 1*66, 2*33, 3*22, 4*14, 6*11 1, 2, 3, 4, 6, 11, 14, 22, 33, 66
13. 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: 6Wait… there is another way…. A little bit easier too
14. 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. 15. PRIME FACTORING Another way to do factoring is to use prime numbers1. Divide the integer by 2, then continue doing so until you can not continue2. Move to the next prime number and just like before keep dividing until you cannot, than move to the next prime number and repeat3. You can stop when your Quotient is prime Tip… As you go circle all the prime numbers
16. 16. PRIME FACTORIZATION OF 100 100 2 is the first prime so 2 X 50 we start with byRemember to dividing 100 by 2circle the primenumbers. 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. 17. EXAMPLES10 18 25 2012 2555/ / / / / 2 5 2 9 5 5 2 1006 5 511 / / / 3 3 2 505 7 73 / 5 101
18. 18. EXAMPLES 10 18 25 2020 2555 The Prime Factors are:/ / / / / 10 2*52 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. 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. 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. 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. 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. 23. FACTORINGWI T H V A R I A B L E S
24. 24. FACTORING A VARIABLE How many y’s? Factorsy 1 yy2 2 y*yy3 3 y*y*yy4 4 y*y*y*yy5 5 y*y*y*y*yy6 6 y*y*y*y*y*yy7 7 y*y*y*y*y*y*yy8 8 y*y*y*y*y*y*y*y
25. 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. 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*y32x2y2 2*2*2*2*2 x*x*y*y4x6y5z3 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. 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. 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