2. Introduction A binomial expansion is the expansion of a repeated product or power of a binomial expression. "Binomial" simply means "two terms." We came up with the general rule for expanding binomials, in particular squaring the sum and the difference of two terms : This general rule cancelled out several steps. Those various steps included multiplication, addition and subtraction. One can Imagine that it can get more complex with even bigger numbers and decimals.
3. If you were an engineer 100 years ago, explain how our method may have been useful rather than just using long multiplication? In an engineering profession I would believe that it mainly deals with big numbers, which may not be whole numbers. This means that the binomial expansion can be handy and very useful to minimize the number of mathematical steps needed. Clearly, doing this by direct multiplication gets quite tedious and can be rather difficult for larger powers or more complicated expressions. (a + b)0 = 1 (a + b)1 = a + b (a + b)2 = a2 + 2ab + b2 (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5
4. To quickly expand a binomial raised to a power which saves a lot of arithmetic, thus reducing the likelihood of making errors. For an engineer this is extremely important as an engineers work needs to be extremely precise. Engineers can use the binomial theory for forecasting, because if you want to build a compound with “n” number of villas each with the same dimension then you can easily square 2 , cube 3 etc….
5. Ex. At what point would our method be big and cumbersome? 123.56(yd) - - - - - - - - - - - - - - ) = ( )( SOCCER FIELD 79.09 (yd) - - - - - - - - - - - - - - - - - - - - - - - - - - - - FENCE (123.6 + 79.9) = 123.56 + 2 x 123.56 x 79.09 + 79.09 It sometimes can get very complicated to use the binomial expansion method when you have big numbers and two or more decimal places. As shown in the diagram above when trying to measure the perimeter of two soccer fields with strenuous numbers and decimal places it becomes very complicated and you have to think twice before using this method.
6. Can you give us some detailed explanations and examples of where long multiplication is more efficient than out expansion method. Long multiplication can be much more efficient than by the binomial expansion method is when either (a or b) is zero. Ex. (a + 0 ) 2 = ( a+ 0) (a + 0 ) = a2+ a0+ a0+02 = a2+ 2(0)a+0
7. Conclusion To quickly expand a binomial raised to a power which saves a lot of arithmetic, thus reducing the likelihood of making errors. For engineers and doctors this is great because they don’t want any mistakes that can threaten someone’s life. This is also a tool they can use to forecast. It sometimes can get very complicated to use the binomial expansion method when you have big numbers and two or more decimal places.
