Binomial Expansion

By: Ghaida Odah 8C

IntroductionDuring an Investigation we came up

with a general rule for binomial expansion. It was (a+b) = a2 + 2ab + b2. We came up with this method

so that it is easy to expans binomials because all we have to do

is substitut ethe variable.

Use of our Method in Engineering Long Ago A hundred years ago the calculation of big numbers was extremely difficult because not many methods were developed. People didn’t have calculators but instead everything was done with pencil and paper. For engineers it was particularly hard because they needed to take a lot of measurements and a lot of multiplication took place. The method that we came up with would have helped them a lot because in some cases engineers would have had to at some point multiply big numbers in order to figure out the area of something and with long multiplication some mistakes could occur which could lead to a lot of problems when it comes to building a house or or anything because if the measurements are not correct than the building will be out of proportion.

If our method was used then engineers wouldn’t have had to multiply a lot of big numbers together which means that the amount of errors possible would decrease resulting in completing the measurements faster and creating good buildings.

When Does Our Method Start to Get Cumbersome?

The method we came up with is very useful but it gets to a point where using other methods that might take longer actually work better. Binomial expansion can work reasonably well up to a power of eight, nine or ten because then it gets too much and a lot of mistakes can occur when you are multiplying all these brackets together. For example of we get an equation like (2x -5y) 8

it will be pretty hard to solve, basically you are going to have (2x-5y) multiplied by itself 8 times and that takes a lot of times and a lot of mistakes can occur.

  This method is also not very effective in other situations. As the number gets closer to decimals it harder to multiply by because decimals aren’t always easy to multiply in your head and the reason we are using binomial expansion is because we need to multiply small numbers on our head then we add the products together rather than just multiplying big number using a calculator or pencil an paper. For example you cannot multiply by 0.00237 in your head but you can do it with 0.01 or by 0.12. When the number gets to more than 2 decimal places it is very hard to work with.

It’s not only about the numbers being too small but also when the numbers are too big and especially when they are odd it gets too hard. For example if you get a number like 980 even though it is an even number it’s still hard to expand that to 2 small numbers that are easy to multiply with though it still works better than odd number because you can break down 980 to 900+80 and then you can multiply without the zeros and you can add them later on but if the number was 981 you are going to end up multiplying by a big number like 81 which is extremely difficult to do in your head.

This is 9802 using the method we came up with: (980)2 = ( 900+80) (900+80) = 9002 + 2(80 x 900) + 802

= 810000 +144000 + 6400 = 960, 400Now we were able to use binomial expansion for this question even though the number was pretty big because it is even so we were able to multiply using these numbers in our heads because there were a lot of zeros which is the case with most even numbers.

This is 9812 using our method we came up with:

(981)2 = ( 900+81) (900+81) = 9002 + 2(900x81) + 812

= 810000 + 145, 800 + 6561 = 962, 316 Even though we managed to get the answer there was no way that this could have been done without a calculator or pencil and paper because nobody can multiply 900 and 81 in their heads and no bodycan square 81 in their head which is the whole point of binomial expansion.

When our method doesn’t work and starts to become big and cumbersome we need to start using other methods. One of these other methods is long multiplication. Even though it might take longer when it comes to multiplying decimals and big odd numbers it works really good. This is because when you multiply decimals using long multiplication everything is laid out and you can use different methods like moving the decimals until it’s a whole number and then adding the decimals at the end. This method is very hard to apply when using binomial expanding. Also when we multiply big numbers especially when they are odd it is much more efficient because the numbers are organized under each other and you can easily check your answer and see where you went wrong if any mistakes occurred. For example if we had to find 2.8342 this is what it would look like if we had to used the method we came up with which is (a+b) 2 = a2+2ab+b2

(2.834)2 = ( 2+0.834) (2 +0.834) = 22 + 2 x 2 x0.834 +0.8342

= 4 + 3.336 + 0.69555 = 8.03155

 

When is Long Multiplication more Efficient than Our Method?

As you can see it is very hard to multiply with 0.834 because it’s a decimal and it has more than 2 decimal places. But when we used long multiplication even though it was quite long it was still easier because it was like multiplying two whole numbers because we added the decimals in the end but you cannot do that with binomial expansion because its all done in your head. Long Multiplication Method:

   

       

Also long multiplication can be used to find different quantities such as speed, time and height etc. If we use binomial expansion we cannot figure out these relationships. Also long multiplication is used in a lot of everyday things such as in science, in medicine and also in the economy while the method of binomial expansion that wee came up with isn’t used as much in these forms. Long multiplication is also used in things like measuring the volume of things like water tanks, and swimming pools so it is very effective in different ways.

In the end we know that the method we came up with is useful not only in mathematics but also in

real-life situations. But, at times and in some situations the method doesn’t work very well and at that point we have to use long multiplication

especially in a lot of real life situations.

OR (a+b)2 = a2 + 2ab + b2