Binomial Expansion Reflection

1. Binomial Expansion Reflection By: Assma Shabab 8D

2. In our investigation that we did we looked at: (0.99)2 = (1- 0.01) (1- 0.01) =12- 2 x 1 x 0.01 + 0.012 Recap

3. We came up with the general rule for expanding binomials, in particular squaring the sum and difference of two terms: (a+b)2 = a2 + 2ab + b2 and (a- b)2 = a2 – 2ab + b2

4. We now have calculators that perform complex numerical calculations easily, but in the bad old days it was all done with pencil and paper. To help them would it be better to use the long multiplication method or the binomial expansion method? Main Idea for Reflection

5. If you were an engineer 100 years ago, explain how our method may have been useful rather than just using long multiplication? Question #1

6. Our method of binomial expansion would be very useful, because if you just did long multiplication (especially with big numbers) it would get too complicated and the answer would not be that accurate (because it is possible that you might get confused by keeping track of the numbers so you might make a mistake which affects the whole equation). But if you used binomial expansion it would make things way much more easier, and simpler to solve. Also the binomial expansion is very satisfying, it makes mathematics easy and enjoyable and encourages people to do it.But it all depends on what number it is, because if there is too much decimal places then it might get complicated to use our method or be complicated to use the long multiplication method. Answer

7. Binomial Expansion: 952= (100–5) (100–5) = 10,000 – 1,000 + 25 = 9,025 Long Multiplication: 952= 95 x95 475 855 9,025 We first say 100 – 5 because it is a way for our method to be effective instead of just saying 95. We put 100 – 5 twice because it is powered by 2. Then we “foil”, and simplify, then we do the equation and get the answer. Simple as that! In the long multiplication we multiply the number by itself which is 95. So we do 95 x95, we do it by using the long multiplication method, which takes a long time, and sometimes gets complicating. For Example:

8. At what point would our method be big and cumbersome? (ie. How many decimal places or what sorts of numbers would make us think twice about using this method.) Question #2

9. In my opinion, our method can be complicated if there are many decimal places (beyond 4 decimal places) or if the exponent of the number is increased then it will be too big and cumbersome to solve. Answer

10. Binomial Expansion – adding one more decimal place: 0.9992= (1 – 0.001) (1 – 0.001) =12 – 0.001 – 0.001 + 0.000001 =12 + 0.000001 =0.000001 0.99992= (1 – 0.0001) (1 – 0.0001) = 12 – 0.0001 – 0.0001 + 0.00000001 = 12 + 0.00000001 = 0.00000001 0.999992= (1 – 0.00001) (1 – 0.00001) =12 + 0.00001 – 0.00001 + 0.0000000001 =12 + 0.0000000001 =0.0000000001 In this example all that I did was add one more decimal place to the number that I used, which was 0.99. I did this for three times. It wasn’t that complicated, except the fact that there were too many zeros involved in solving this problem which made it a bit confusing. If it is confusing to me then for the people 100 years ago it would be a bit more confusing to them. For Example:

11. Binomial Expansion – increasing the exponent: 0.992= (1 – 0.01) (1 – 0.01) =12 – 0.01 – 0.01 + 0.0001 =12 + 0.0001 = 0.0001 0.993= (1 – 0.01) (1 – 0.01)(1 – 0.01) =13 – 0.01 – 0.01 – 0.01 + 0.0001 + 0.01 + 0.0001 =13 + 0.0002 + 0.01 =0.970299 0.994= (1 – 0.01) (1 – 0.01)(1 – 0.01)(1 – 0.01) =14 – 0.01 – 0.01 – 0.01 + 0.0001 + 0.0001 + 0.0001 = 14 + 0.0003 = 0.96059601 In this example all I did was increase the number of the exponent. So now instead of having 2 brackets I now have more than 2. When I increased the number of exponents the way to solve it got very complicated and was hard to do using pencil and paper, but if I used a calculator it was way much more easier (I used it to find the answer) Now is the point when the numbers are too big and cumbersome to work and solve with.

12. Can you give us some detailed explanations and examples of where long multiplication is more efficient than our expansion method? Question #3

13. There are some occasions when long multiplication is a better method then using binomial expansion. From my research this only happens if you have 2 numbers to multiply with. It’s easy because it is straight forward and doesn’t need a lot of effort as much as the binomial expansion needs. Because in the binomial you need to times this with this, then this with that, and it takes a lot of time and thinking. But the long multiplication its very simple and easy to solve, and doesn’t need a lot of effort. Answer

14. Binomial Expansion: 992= (100 – 1) (100 – 1) =10,000 – 100 – 100 + 1 =10,000 – 200 + 1 =9,801 Long Multiplication: 992=99 x99 891 891 9801 In the first example it shows us a binomial version of how to solve 992, which takes a lot of time and effort than just multiplying it the long way. The long way is more simpler and straight forward. For Example:

15. In the end I think that Binomial Expansion has some positive and negative sides. It can save you lots of time and it can waste it. It can waste time because (especially if the number is powered by 2) you have to do all the steps of the method, and it takes time instead of just doing long multiplication. I also think that binomial expansion can help a lot in the future unlike the past. Because in the future I predict that it will be so busy and people are not bothered to waste their time in just solving 994 so they will refer to the binomial method and it will help save time a a lot. Conclusion

16. The End!!!(please give me a good grade )