BINOMIAL EXPANSIONS - Reflection

1. BINOMIAL EXPANSIONS - Reflection By: Dana Asaad Class: 8B Tuesday, November 2, 2010

2. Introduction It is rather customary to perform complicated mathematical computations by using some short cut methods. These short cuts can be some identities. The beauty of it is when it comes to how often one implements it in mathematical applications. Consider a simple calculation like 95² = 95x95 An average student will calculate it by multiplying it itself twice as follows. 95 x 95 ----- 475 8550 ------ 9025 ------ But an above average student may try to do it in a short cut manner rather mentally 95 = (100-5) So 95 x 95 = (100-5) x (100-5) = 100 x 100 + 5 x 5 – 2*100*5 = 10000 + 25 – 1000 = 9025

3. More Examples Consider another example: 70.5² 70.5 x 70.5 We will apply our new method we adopted in this case. 70.5 x 70.5 = (70+0.5) x (70+0.5) = 70 * 70 + 0.5*0.5 + 2*0.5*70 = 4900 + 0.25 + 70 = 4970.25

4. Observation Definitely the method above is very useful in our computations. But it does not mean that it is worth to apply in all such cases. Consider the example: (94.58)² (94.58)² = (94 + 0.58)² = 94² + 0.58² + 2 x 94 x 0.58 = 8836 + 0.3364 + 109.04 = 8945.3764

5. Comments The expansion method mentioned above is not appropriate for our new method which is another way used for solving binomial expansions because the numbers are very large and includes decimal points which makes it extremely difficult calculate mentally. Here our usual long multiplication method is more efficient than our expansion method. Even then, our expansion method can be applied where ever possible.

6. Example Consider another example where we can easily apply another expansion method a² - b² = (a + b) ( a – b) Compute 107 x 93 = (100 + 7) (100 – 7) = 100² - 7² = 10000 – 49 = 9951 Here our expansion method is very efficient because we can calculate the result mentally by applying the expansion method.

7. 100 years ago Now, in the modern world we have powerful calculators which can perform complex calculations very easily, but that was not the case a few decades ago. In those days people had to use a pencil and a paper to work out many difficult binomial expansions by hand. Think about an engineer and how long it must have taken him to work out something that would probably take us less than a minute to work out using our modern technology. Since they didn’t have calculators to help them work out their answers, this new technique would be useful to them because they could calculate an binomial expansion more quickly and also more efficiently because if you think about it your really just breaking the numbers down to simpler ones in order to work it out mentally and easily.

8. Conclusion So to conclude this, I would like to state that there are many short cut methods to save you a lot of time and confusion. Each of these expansion methods are all excellent but some work better than others in different cases. For example, as I mentioned before decimal numbers more than three decimal places can get a little tricky while using the new bionomic expansion method so therefore the long multiplication method would be more efficient in order to come up with an answer, but if you were given a whole number or a number divisible by, lets say ten you would most likely use our new method because it would probably take you half the time to answer and there would be less working out than with the long multiplication and so there is a bigger chance that your answer will be correct.