Binomial Expansions - Reflection

1. Binomial Expansions - Reflection By : Aya Ibrahim 8A

2. What will this be about? This reflection will be about binomial expansions, how they helped us in the past, and how they help us now. We have currently figured an expansion method to make squaring numbers and variables easier. But are they all that easy? Will they work with all kinds of sums? Or will they work with only a few? We’ll also find out whether you can use this method with squares, or even cubes without it getting long and cumbersome. However, if it does, will we all turn back to long multiplication, if not calculators?

3. Question 1) • If you were an engineer 100 years ago, would our method be more useful than using long multiplication? • Engineers, of any time, use math everyday. They’re always working with spaces and area or perimeter. Our method is extremely useful in cases of finding area. Let’s say that 100 years ago, an engineer wants to build a school. The man he is working for has a piece of land that is 10,000 meters squared. He gives tells the engineer that the area should be divided into 4 unequal pieces, with the biggest being the classrooms, the second and third biggest (being of equal size) for the library, and the remaining area to become a teacher’s staff room. The engineer comes up with a plan that looks like this:

5. In the top left-hand corner of that screen-shot, is our generalized formula. In this case, the total area is going to be 10,000m2. How? In order to get the area, you must multiply length by width. • 80x20= 100 • 80x20=100 • 100x100=10,000. • AxB to the power of 2 is the area of this piece of land.

6. Question 2) • At what point would our method be big and cumbersome? • Now, as you can see, this method works well with most engineer projects. What about bigger problems? Will more decimal places, numbers and indices make it harder for us to use this method? • The answer is, “yes”. This method becomes really big, long and cumbersome when the calculating the surface area of minute objects such as medical instruments or tiny gadget microchips. It also becomes a challenge when the last digit of a two-digit number is NOT zero. That being said, if the numbers being used are under 10, or end with zero, this method comes to mind.

7. Question 3) • What are some examples and explanations that claim that long multiplication is more efficient than our expansion method?  • As stated above, it becomes harder for us to use the expansion method with two-digit numbers that don’t end with a zero. That’s where long multiplication comes in handy. In cases of school, long multiplication can’t be used in algebra, because algebra is where we use the expansion method, or the distributive law to figure out answers. Another thing is that long multiplication uses no variables. When trying to calculate decimals, I personally prefer, either: • a) Using a calculator • Or, • b) Using Long Multiplication • Here’s an example of how it’s easier to use. Lets take a look at how we can do 82x82 in long multiplication and how to do it using the expansion method. • Long Multiplication:

9. Expansion Method: • (82)to the power of 2 • (82) x (82) • = 6724 • Now, if you were to use this method on the test, you would eventually turn to long multiplication to figure out the product of 82 multiplied by 82. In the expansion method, it SEEMS like there is no working our, even if you did it in your head, which could spark problems with SOME tests.