1. Binomial Expansions - Reflection By: Salman Al-Sulaiti

2. Rule Squaring the sum of two terms (a+b) 2 = a 2 + 2ab + b 2 Squaring the difference of two terms (a-b) 2 = a 2 – 2ab +b 2

3. Usefulness If you were an engineer 100 years ago, explain how our method may have been useful rather than just using long multiplication? Using this method is more time efficient allowing you to finish more in less time. Using this method is more time efficient allowing you to finish more in less time. This method is also another way to square numbers by converting them into the sum/difference of two numbers. This method is also another way to square numbers by converting them into the sum/difference of two numbers. Easily multiply numbers close to 10, 100, …… Easily multiply numbers close to 10, 100, ……

4. Comparison Binomial Expansion Long Multiplication 101 2 = (100+1) 2 101 2 = (100+1) 2 = 100 2 + 2x100x1 + 1 2 = 100 2 + 2x100x1 + 1 2 = 10000 + 200 + 1 = 10000 + 200 + 1 = 10201 = 10201 101 2 = 101 x101 x101 101 101 0000 0000 +10100 +10100 10201 10201

5. Comparison Explanation The binomial expansion was easy because the numbers used were all close to 100. It was also easily lain out for you and made the multiplication and addition easier and although the answer was a 5 digit number the working out was easy because most of the numbers were 0.

6. Downside At what point does the method become big and cumbersome? When the number has many digits (Not zeros) or/and many decimal places were the number of places becomes too much to do mentally.

7. Example (69.5046) 2 = (70-1+0.5+0.004+0.0006) 2 = 70 2 - 70 + 35 + 0.28 + 0.042 – 70 + 1 2 – 0.5 – 0.004 – 0.0006 + 35 – 0.5 + 0.5 2 + 0.002 + 0.0003 + 0.28 – 0.004 + 0.002 + 0.004 2 + 0.0000024 + 0.042 – 0.0006 + 0.0003 + 0.0000024 + 0.0006 = 4830.88942116

8. Limitations Although this method works well for most things it does have its limitations where long multiplication could be used as a better alternative for example : When the binomial method is used with numbers that are cubed or powers larger than that When multiplying many two or more digit numbers In situations where the number has to be broken up into more than just a and b (more than two numbers)

9. Limitations Explanation When numbers start to become 3 digits or even 4 the method becomes harder and you will then have to do complex addition and will begin to use long multiplication within the expansion which will beat the purpose of the method. Also when these numbers are split up into more than two parts the rule will not work and you will end up adding and multiplying every number separately which will just mean your doing long multiplication in a more complicated way by just adding expansion to it.

10. Examples 23 3 = (20+3) (20+3) (20+3) = 20 2 + 2x20x3 + 520x20 + 520x3 = 20 2 + 2x20x3 + 520x20 + 520x3 = 12167 = 12167 (13) (15) (21) = (20+1) (10+5) (10+3) = 20x10 + 20x5 + 1x10 + 1x5 + 315x10 + 315x3 = 20x10 + 20x5 + 1x10 + 1x5 + 315x10 + 315x3 = 4095 = 4095 311 2 = (300+10+1) (300+10+1) = 300 2 + 300x10 + 300x1 + 10x300 + 10 2 + 10x1 + 1x300 + 1x10 + 1 2 = 300 2 + 300x10 + 300x1 + 10x300 + 10 2 + 10x1 + 1x300 + 1x10 + 1 2 = 96721 = 96721

11. Conclusion Overall we found out that binomial expansion is an efficient way to multiply numbers and square them. It’s a shortcut way instead of long multiplication but still in some things long multiplication can’t be completely abandoned because expansion becomes confusing and inefficient when dealing with decimals, large numbers, and large powers (over squaring).