Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

Binomial Expansion and Surds. Part 1 By Mr Porter Assumed Knowledge. Student should be able to expand a single bracket expression. Examples: Expand (and simplify) ( a) 2 3 - 5 ( ) = 2 ´ 3- 5 The meaning of bracket is X (times) ) Each item in the bracket is to multiplied by the item on the outside , in this case ‘2’. = 2 ´ 3- 2 ´ 5 The ‘number’ part can be simplified, the √5 behaves like algebra. = 6-2 5 This is the final answer. Note : Number multiply number and surds multiply surds, but number and surds, behave like algebra. 1 {Assumed Knowledge.} Examples: Expand ( b) 2 5 + 2 ( ) = 2 ´ 5+ 2 The meaning of bracket is X (times) ) Each item in the bracket is to multiplied by the item on the outside , in this case ‘√2’. = 2 ´5- 2 ´ 2 The ‘surd’ part can be simplified, the (number/surd) behaves like algebra. =5 2 -2 This is the final answer. You should know √2 X √2 = √4 = 2 or (√2)2 = 2 Note : Number multiply number and surds multiply surds, but number and surds, behave like algebra. 2 {Assumed Knowledge.} Examples: Expand c) ( 2 3- 5 2 ( ) = 2 ´ 3- 5 2 The meaning of bracket is X (times) ) Each item in the bracket is to multiplied by the item on the outside , in this case ‘√2’. = 2 ´ 3- 2 ´ 5 2 The ‘surd’ part can be simplified, the √2 x 3 behaves like algebra. = 3 2 -5´2 This is the final answer. You should know √2 x 5√2 = 5√4 =5 x 2 or (√2)2 = 2 = 3 2 -10 This is the final answer. Note : Numbers multiply numbers and surds multiply surds, but number and surds, behave like algebra. 2 Binomial Product and surds. This is like combining any two of the examples of single brackets. Examples: Expand (and simplify) ( 3 + 5 )( 4 + 2 ) Binomial, meaning 2 brackets in this case to be expanded, we use the distributive law. (There are other methods.) To apply the distributive law, split one of the brackets. [Usually, one containing a ‘+’ sign] Then , multiply the second bracket by each of the parts of the bracket you split. ( ) ( = 3´ 4 + 2 + 5 ´ 4 + 2 ) Now, multiply out each of the individual brackets. = 3´ 4 + 3 ´ 2 + 5 ´ 4 + 5 ´ 2 = 12 + 3 2 + 4 5 + 10 This expansion is finished. Numbers multiply numbers and surds multiply surds, but numbers and surds, behave like algebra. Now, look very HARD at the surd parts. All the surds are different (just like algebra) and they cannot be broken down by the square numbers 4, 9, 16, 25, …. Binomial Product and surds. Examples: Expand (and simplify) ( )( 2+ 2 7- 2 ( ) ) Last example was very boring, but necessary. To apply the distributive law, split one of the brackets. [Usually, one containing a ‘+’ sign] Then , multiply the second bracket by each of the parts of the bracket you split. ( = 2´ 7- 2 + 2 ´ 7- 2 ) Notice, the un-split bracket is written out twice. Now, multiply out each of the individual brackets. = 2´7-2´ 2+ 2 ´7- 2 ´ 2 = 14 - 2 2 + 7 2 - 2 Numbers multiply numbers and surds multiply surds, but numbers and surds, behave like algebra. Did you remember that √2 x √2 = √4 = 2 or (√2)2 = 2 Now, look very HARD at the surd parts. All the surds are the same (just like algebra) and they can be combined. = 12 + 5 2 This expansion is finished. Binomial Product and surds are more interesting, when the surd part is the same! Binomial Product and surds. Examples: Expand (and simplify) To apply the distributive law, split one of the brackets. [Usually, one containing a ‘+’ sign] Otherwise, the bracket with the simplest surd. Then , multiply the second bracket by each of the parts of the bracket you split. ( 5 - 3 )(1- 4 3 ) ( = 5 ´ 1- 4 3 ) ( - 3 ´ 1- 4 3 ) Notice, the un-split bracket is written out twice. Now, multiply out each of the individual brackets (take care of negative signs!). = 5 ´1- 5 ´ 4 3 - 3 ´1- - 3 ´ 4 3 = 5 - 20 3 -1 3 + 4 ´ 3 DOUBLE negative is ‘+’ Did you remember that √3 x √3 = √9 = 3 or (√3)2 = 3 Now, look very HARD at the surd parts. All the surds are the same (just like algebra) and they can be combined. = 17 - 21 3 This expansion is finished. Binomial Product and surds are more interesting, when the surd part is the same! Binomial Product and surds. Examples: Expand (and simplify) (2 5- 3 )( ( 5+4 3 = 5´ 2 5- 3 ) ) To apply the distributive law, split one of the brackets. [Usually, one containing a ‘+’ sign] Otherwise, the bracket with the simplest surd. Then , multiply the second bracket by each of the parts of the bracket you split. ( +4 3´ 2 5 - 3 = 2 ´ 5 - 15 + 8 15 - 4 ´ 3 ) Notice, the un-split bracket is written out twice. Now, multiply out each of the individual brackets (take care of negative signs!). Did you remember that √5 x √5 = √25 = 5 or (√5)2 = 5 = 10 - 15 + 8 15 +12 √3 x √3 = √9 = 3 or (√3)2 = 3 Now, look very HARD at the surd parts. All the surds are the same (just like algebra) and they cannot be broken down by the square numbers 4, 9, 16, 25, …. = 22 + 7 15 This expansion is finished. Binomial Product and surds are more interesting, when the surd part is the same! Binomial Product and surds. Examples: Expand (and simplify) ( ) ( 2 5-3 = = 5´ ( )( 5-3 ) 5-3 - 3´ To apply the distributive law, split one of the brackets. [Usually, one containing a ‘+’ sign] Otherwise, the bracket with the simplest surd. Then , multiply the second bracket by each of the parts of the bracket you split. ) 5-3 ( = 5 - 3 5 - 3 5 + 3´ 3 (….)2 = (…..) (…..), the first step is the remove the power ‘2’. ) 5-3 Notice, the un-split bracket is written out twice. Now, multiply out each of the individual brackets (take care of negative signs!). Did you remember that √5 x √5 = √25 = 5 or (√5)2 = 5 =5 -3 5-3 5 +9 = 14 - 6 5 Now, look very HARD at the surd parts. All the surds are the same (just like algebra) and they cannot be broken down by the square numbers 4, 9, 16, 25, …. This expansion is finished. Binomial Product and surds are more interesting, when the surd part is the same! Binomial Product and surds. Examples: Expand (and simplify) (2 ) ( )( 2 5+ 3 = 2 5+ 3 2 5+ 3 ( =2 5´ 2 5+ 3 ( = 2 5 ) 2 ) To apply the distributive law, split one of the brackets. [Usually, one containing a ‘+’ sign] Otherwise, the bracket with the simplest surd. Then , multiply the second bracket by each of the parts of the bracket you split. ) + 3´ 2 5 + 3 + 2 15 + 2 15 + = 20 + 2 15 + 2 15 + 3 = 23 + 4 15 ( (….)2 = (…..) (…..), the first step is the remove the power ‘2’. ( 3) 2 ) Notice, the un-split bracket is written out twice. Now, multiply out each of the individual brackets (take care of negative signs!). Did you remember that 2√5 x 2√5 = 4√25 = 20 or (2√5)2 = 4 x 5 = 20 √3 x √3 = √9 = 3 or (√3)2 = 3 Now, look very HARD at the surd parts. All the surds are the same (just like algebra) and they cannot be broken down by the square numbers 4, 9, 16, 25, …. This expansion is finished. Binomial Product and surds are more interesting, when the surd part is the same!