Binomial Expansion ReflectionAkar Khoschnau, Grade 8B

Information

The Question was

“is there an easy way to do 0.992?”

Well guess what ? We can, by binomial expansions

Check the example below:

0.992= (1-0.01) (1-0.01)

= 12-2x1-0.01+(-0.01) (-0.01)

= 0.9801

This way you can see that the method we were taught in class can really help us finish expansions like these really quickly rather than just doing long multiplication which wastes time if you do in a test.

Benefits of normal multiplications

If you were an engineer 100 years ago, explain how our method may have been useful rather than just long multiplication?

well An engineer 100 years ago could use this method as a shorter way to solve this. Imagine your self a 100 years ago, you wouldn’t have anything except paper to write on.

So really using a shorter way can gain you more time, and who knows, the engineer might have developed a new way to solve more problems

This Method is really important for people like mathematicians or engineers, because they have a specific time for solving something.

Engineer using binomial expansion at work

Well if an engineer uses the method we use, then he can really build a house really fast, like measuring the area with the method, or discovering the length or width rather than using long multiplication that will take him a little longer to figure out. For someone like an engineer, this method is: Useful, because it is very easy, and has many apps Trustable because it finishes really fast and takes a little space. Because it has a lower chance of error, it is accurate, and in

engineering like building houses, it needs to be, because people

are depending on this

Rules

For the square of the sum of two number we have found… check example below:

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

This means the first term squared, plus the product of the first and second term times two, plus the last term squared which describes the equation

But the square of difference of two numbers we developed:

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

This is a bit different, where the first term is still squared, however the product of the first and second term squared is then subtracted from the first term squared, then the last term squared is added so it explains the equation.

Explain

This method can give you two answers, one for adding and one for subtracting

You use numbers that can be really easy to do.

writing the number as a sum or difference you will have the same result, but during the equation the multiplication is a bit different.

Binomial expansion vs multiplication

Binomial Expansion

992= (100-1)2

= 10000 – 2x100x1 + 1

= 9801

Faster and took smaller space

Multiplication

992= 899 x99 891

+8910longer took space.

9801

Not always a good idea

In some equations, our method becomes hard and messy, When the equation has lots of decimal places

and it has to go to the left, then its will become messy.

Sometimes there are decimals with numbers such as 1.3203 or 0.001410 which will really confuse you.

Examples

83.42= (80+3+0.4)2

= 802+80x3+80x0.4+3x80+32 +3x0.4+0.4x80+0.4x3+ 0.42

= 6400+240+32+240+9+0.12+32+0.12+0.16

= 6955.56

83.4x83.43336

2502066720

0

6955.56

Sometimes long multiplication is better than using binomial expansions

Sometimes using long multiplication is better than using our method number which have squared or cubed etc.. Numbers that have 3 or more digits

Limitations Explained

When the equation starts to have numbers 3,4 or 5 digits, our method becomes complicated, because it will involve multiplications, and you will probably end up using long multiplication to calculate within the original calculation, so you get a better more accurate answer.

Example

523= (50+2) (50+2) (50+2)

= 503+50x2x2+2x50+2x50+2x2x2

= 125000+200+100+100+8

= 140608

52x52104

26002704 x525408

135200

140608

At the tip of Pascal's Triangle is the number 1, which makes up the zero row. The first row (1 & 1) contains two 1's, both formed by adding the two numbers above them to the left and the right, in this case 1 and 0 (all numbers outside the Triangle are 0's). Do the same to create the 2nd row: 0+1=1; 1+1=2; 1+0=1. And the third: 0+1=1; 1+2=3; 2+1=3; 1+0=1. In this way, the rows of the triangle go on infinity.

Pascal’s Triangle

Conclusion

well, we have learned that Binomial expansions are very useful and very easy to do, it takes really a small amount of time to finish with 2 to 3 digit numbers.

It’s a good way to use the method in times where you have a small amount of time to finish something, like in a test, but this method cannot really replace long multiplication, as I mentioned, when it comes to 4 -5 etc.. Digit numbers, its really confusing, so using long multiplication in that case is a lot better.