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Name_____________________________________________Date_________Period __________ Greatest Common Factor (GCF) Definition: In a multiplication expression, the quantities being multiplied are called the FACTORS. Example: 42 6 7 Example: 3x 3 x therefore 6 and 7 are FACTORS of 42. therefore 3 and x are the FACTORS of 3x . A number or expression can be factored into lowest terms. This means taking the non-prime factors to the lowest number possible. Example 1: 42 6 7 3 2 7 Therefore 3 2 7 is considered lowest terms because 3, 2, and 7 are all prime numbers and cannot be simplified any lower. the factored form of 42 is 3 2 7 . Example 2: Factor 56 into lowest terms. (Sometimes a factor tree helps!!!) 56 56 7 8 7 8 56 7 2 4 56 7 2 2 2 Final Answer 2 4 2 2 Example 3: Factor 2x y into lowest terms. 3 2x 3 y 2 x 3 y 2x3y = 2 x x x y Final Answer If the number does not have any factors other than itself and 1, then the number is said to be PRIME. Example: Factor 13 into lowest terms. 13 13 1 Therefore 13 is prime. Factor the following expressions to lowest terms. 1. 33 ___________________ 2. 40 _______________________ 3. 100 ___________________ 4 4. a b _______________________ 2 2 5. 50n m ___________________ Find the missing factor. 6. y 5 ?y 3 2 8. 7x ? 7x _________ 7 3 7. c ? c _________ 9. 6p 5 ? 6p 4 _________ _________ The GREATEST COMMON FACTOR (GCF) is the greatest number that is a factor of all the numbers. Example 4: Find the GCF of 21 and 15. 21 3 7 15 3 5 3 is the biggest number that they have in COMMON therefore the GCF is 3. Example 5: Find the GCF of w2z4 and u2w4. w 2z 4 w w z z z z 2 2 The common terms are w w w , therefore w is the GCF. u 2w 4 u u w w w w Example 6: Find the GCF of 18 and 24. 18 2 3 3 24 2 2 2 3 Each term has a 2 & a 3 in common therefore the GCF is 2 3 6 Find the greatest common factor. 10. 28, 42 __________ 12. 75a3, 125a2 __________ 11. 55uv2, 65u2w2 __________ 13. 120a3b2, 90ab3 __________ When factoring polynomials, the first step is to find the GCF between the 2 terms. Example 7: 5a + 35 GCF of 5a and 35 is 5 Once you have the GCF, the GCF gets “taken out” of the expression and the factors that are “left” stay in parentheses as an expression. GCF 5•a 5•7 5 (a+7) “left over” from 1st term therefore 5(a + 7) is the solution “left over” from 2nd term A sure way of knowing if your answer is correct is to use the distributive property. 5(a + 7) = 5a + 35 **Factoring polynomials is just like doing the distributive property backwards!! Example 8: 4x + 6y GCF = Check: ___________________ Example 9: -42gh - 30h2 GFC = Check: __________________ 3 4 5 3 4 2 2 2 Example 10: 42a b x 56a x 168a b x GCF = Factor. 14. -6c2 - 3 15. 4k3r2q - k2rq2 - 10kq 16. 108c2n2z5 - 35c4nz4 17. 3k – 12k2g + 30k5 18. Barbara has a rectangular lettuce garden. The area of the garden is (25x2 + 15x – 5x3). What is the length if the width is (5x + 3 – x2)? 19. Jessica was paid $(-6x2p2 + 30xp5 – 15x5p3) for working a week at Kroger. If she was paid an hourly rate of $(-3xp2) per hour, how many hours did she work?