Reflection: Binomial Expansion

1. Reflection:Binomial Expansion By: Mohamad Tiba 8B

2. Introduction • During this assignment, I have been asked to write about how the binomial expansion method could be used instead of long multiplication. I will be discussing how binomial expansion can solve problems without as many steps and it saves time compared to long multiplication. I will also describe when long multiplication should be used instead of binomial expansion. • Binomial Rule: (a+b)2 = a2 + 2ab + b2 (a-b)2 = a2 - 2ab + b2

3. If you were an engineer 100 years ago, explain how our method would have been useful rather than just using long multiplication. • Engineers always work with large numbers, need to be accurate and are required finish in a certain amount of time. Because of these reasons, it would be easier to use the binomial expansion instead of long multiplication. • Binomial expansion is not confusing and easy to finish. • 100 years ago, they didn’t have calculators and our method would have easier to use while working with large numbers and this would help them to keep to their time limit.

4. At what point would our method be big and cumbersome. • Our method would be big and cumbersome when our exponents are greater than 2. Also, it would become difficult if we were using big numbers and decimals. Binomial Expansion: Long Multiplication: 12002 = (500 + 700)(500 + 700) = 250,000 + 350,000 + 350,000 + 490,000 = 250,000 + 700,000 + 490,000 = 1,440,000 1200 X1200 ______ (12x12) = 144 144 then add the zeros you ignored (4 zeros) = 1,440,000 You Decide!

5. Can you give some detailed explanations and examples of where long multiplication is more efficient than our expansion method. • Long multiplication would be better in the situation where you are solving a problem that has decimals, large numbers or with an exponent larger than two. • An example would be when using larger numbers it would be better to use long multiplication because your only multiplying single digits. If you use binomial expansion you would be multiplying large numbers rather than single digit. Single digits can be done using mental math easily but the larger the number the more difficult it becomes. Example cont………….

6. Example cont……. Binomial Expansion-with these numbers it cannot be done mentally because of the large numbers: Long Multiplication-it is easier because we can do single digit multiplication in our head: (1265) 2 = (1200 + 65) (1200 + 65) = (1200 x 1200) + (1200 x 65) + (65 x 1200) + (65 x 65) = 1,440,000 + 78,000 + 78,000 + 4,225 = 1,600,225 1265 X1265 ________ 6325 75900 253000 + 1265000 _________ 1,600,225

7. Conclusion Binomial expansion is a great way to solve problems in less steps. Long multiplication is useful when you are using big numbers or when you have decimals. If I were an engineer and had to choose either method, it would depend on the number. The larger numbers I would use long multiplication because I can do single digit multiplication in my head. If I was multiplying smaller numbers I would use binomial expansion because I could work out the multiplication in my head without using the single digits, which would make it faster.