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Factorization
- 1. FACTORIZATION & GREATEST COMMON FACTORS FINDING THE PARTS OF A MONOMIAL
- 2. MONOMIALS • A Monomialis 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 LE N U M B E R S
- 4. FACTORS • Factors arethe 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 isthe 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 FINDTHE 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 NUMBER88 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 NUMBER18 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 NUMBER250 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 • Sometimes 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 onefind 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 2circle 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 • Primenumbers 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 OF100 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 PRIMENUMBERS • 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 PRIMENUMBERS • 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 HV 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
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