HFCC Math Lab Beginning Algebra 2 MULTIPLICATION AND DIVISION OF SIGNED NUMBERS PART I: Multiplication of Signed Numbers Rules for Multiplication of Signed Numbers: (These Rules must be memorized.) Rule 1: To multiply two numbers with the same sign, multiply absolute values. The result is always positive. Rule 2: To multiply two numbers with different signs, multiply the absolute values. The result is always negative. We can state the Multiplication Rules in another equivalent way: Rule 1: Positive Negative Positive = Positive Negative = Positive Rule 2: Positive Negative = Negative Negative Positive = Negative Note: Even though + 3= 3, the positive signs are included in some of the examples below for emphasis. Examples: (Rule 1) 1. ( 3) ( 5) =+ 15= 15 The signs are the same, so the answer is positive. 2. ( + 6) ( + 2) =+ 12= 12 The signs are the same, so the answer is positive. 3. ( 6) ( 1.1) =+ 6.6= 6.6 The signs are the same, so the answer is positive.. ( 6) ( 7) =+ 2= 2 The signs are the same, so the answer is positive. 3 5 3 5 15 15 5. =+ =+ = The signs are the same, so the answer is positive. 7 7 28 28 1 1 6. ( ) =+ =+ = 2 The signs are the same, so the answer is positive. 2 2 1 2 Examples: (Rule 2) 1. ( + 3) ( ) = 12 The signs are different, so the answer is negative. 2. ( 6) ( + ) = 2 The signs are different, so the answer is negative. 3. ( + 3) ( 1.2) = 3.6 The signs are different, so the answer is negative. Revised 02/09 1
. ( 3) ( + 0.7) = 2.1 The signs are different, so the answer is negative. 5. (6) ( 3) = 18 The signs are different, so the answer is negative. 6. ( 0.02) (0.) = 0.008 The signs are different, so the answer is negative. 1 1 12 12 7. ( + 12) = = = The signs are different, so the answer is negative. 3 3 1 3 1 3 1 3 3 8. The signs are different, so the answer is negative. = = 5 5 20 When multiplying more than two signed numbers together do the multiplications one at a time from left to right. Examples: (Rules 1 and 2). Parentheses are used to indicate multiplication. 1. ( + 2)( 3)( 5) 2. ( 3)( 2)( + 7) 3. ( 2)( + )( 1) ( 6)( 5) ( + 6)( + 7) ( 8)( 1) 30 2 8. ( 2)( 5)( 6) 5. ( 1)( + 3)( + ) 6. (6)(2)( 1) ( ) ( + 10)( 6) 3 ( + ) (12)( 1) 60 12 12 7. (3)( )( + 2) 8. ( ) 9. ( 2) 2 3 ( 12)( + 2) ( )( ) ( 2)( 2)( 2) 2 16 ( + )( 2) 8 Note: In any multiplication problem: If there are an even number of negative factors, the answer is positive. See examples 1, 2 and 3 above. If there are an odd number of negative factors, the answer is negative. See examples, 5, 6 and 7 above. Revised 02/09 2
PART II: Division of Signed Numbers Rules for Division of Signed Numbers: (These Rules must be memorized.) Rule 1: To divide two numbers with the same sign, divide the absolute values. The result is always positive. Rule 2: To divide two numbers with different signs, divide the absolute values. The result is always negative. We can state the Division Rules in another equivalent way: Rule 1: Positive Negative Positive = Positive Negative = Positive Rule 2: Positive Negative = Negative Negative Positive = Negative Notice that the Rules for Division are the same as the Rules for Multiplication. Examples: 1. ( + 8) ( ) = 2 The signs are different, so the answer is negative. 2. ( 20) ( 5) =+ = The signs are the same, so the answer is positive. 3. ( 0) ( + 8) = 5 The signs are different, so the answer is negative.. ( + 50) ( + 5) =+ 10= 10 The signs are the same, so the answer is positive. 5. ( 1.2) ( 0.3) =+ = The signs are the same, so the answer is positive. 6. ( + 2.8) ( 0.) = 7 The signs are different, so the answer is negative. 7. 18 ( 2) = 9 The signs are different, so the answer is negative. 8. ( 2) 3= 8 The signs are different, so the answer is negative. Revised 02/09 3
Remember that a fraction bar means division. Therefore, we can simplify fractions containing signed numbers by using the Division Rules for signed numbers. Examples: + 2 2 1. = 3 3 because a positive divided by a negative is negative. 2 2 2. = + 3 3 because a negative divided by a positive is negative. 2 2 3. =+ 3 3 because a negative divided by a negative is positive.. 5. 6. 7. 8 8 8 2 = = = + 10 10 10 2 5 12 12 12 6 2 =+ = = 18 18 18 6 3 5 5 5 5 1 = = = 20 20 20 5 5 5 2 8 = = = 3 2 3 2 3 5 15 8. 2 1 6 3 6 3 6 7 6 7 2 = = =+ = 35 7 35 7 35 3 35 5 3 1 5 1 9 5 20 9. 1 1 = = = 3 5 3 5 3 9 27 Note: Be careful: 21 21 = 7 whereas = 7 3 7 Revised 02/09
Exercises: Perform the following multiplications and divisions. 1. ( + 3)( 6) 2. ( 2)( 2)( 2)( 2) 3. ( 3)( + ). ( 2)(5) 5. (5)( 3) 6. ( 6)( 7)( 8) 7. ( + 10) ( 5) 8. 12 6 9. ( 6) 10. ( 1) 2 6 7 11. ( 1) 12. ( 8)( 7) 13. ( 6) (2) 1. 8 ( 2) 15. ( 2) ( 6) 16. 16 ( ) 3 2 5 5 17. 18. 3 6 7 19. ( 1.2) ( 0.) 20. ( 1.5) 3 1 1 21. 5 ( 7) 22. 2 3 23. ( 5)( )( 3)( 2) 2. 2 5 3 7 3 5 2 5 25. 26. 8 3 6 Answers to all problems. Solutions to some problems. 1. 18 2. 16 3. 12. 10 5. 15 6. 336 2 7. 2 8. 2 9. ( 6) = ( 6)( 6) = 36 10. 1 11. 1 Note: There are an odd number of negative factors, so the answer is negative. 12. 56 13. 3 1. 15. 7 16. 17. 3 2 3 3 = 9 1 25 = = 1 18. 3 2 8 8 2 Revised 02/09 5
Answers continued: 19. 3 20. 0.5 1 21 21. 5 ( 7) = ( 7) 21 7 = 1 3 21 1 = = 7 1 1 7 7 1 7 22. 2 = = = 23. 120 Note: There are an even number of negative 3 3 1 3 12 factors, so the answer is positive. 3 2. 25. 26. 2 5 2 5 2 5 10 = + = = 3 7 3 7 3 7 21 3 5 3 5 3 5 15 = = = 8 8 8 32 2 2 5 2 5 2 6 2 6 = = =+ = 3 6 3 6 3 5 3 1 5 5 NOTE: You can get additional instruction and practice by going to the following websites: http://www.themathpage.com/alg/multiply-divide-signed-numbers.htm This website gives rules for multiplying and dividing signed numbers along with many interactive examples. http://www.quia.com/jg/1669.html This website includes flashcards, matching and concentration games that can be used to practice the rules for multiplying and dividing signed numbers. Revised 02/09 6