1. Binomial Expansion Reflection Akar Khoschnau, Grade 8B

2. Information The Question was The Question was “is there an easy way to do 0.99 2 ?” “is there an easy way to do 0.99 2 ?” Well guess what ? We can, by binomial expansions Check the example below: Check the example below: 0.99 2 = (1-0.01) (1-0.01) 0.99 2 = (1-0.01) (1-0.01) = 1 2 -2x1-0.01+(-0.01) (-0.01) = 1 2 -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. 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.

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

4. 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: 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 Useful, because it is very easy, and has many apps Trustable because it finishes really fast and takes a little space. 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 th is 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 th is

5. Rules For the square of the sum of two number we have found… check example below: For the square of the sum of two number we have found… check example below: (a+b) 2 = a 2 + 2ab + b 2 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: But the square of difference of two numbers we developed: (a-b) 2 = a 2 - 2ab + b 2 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.

6. Explain This method can give you two answers, one for adding and one for subtracting This method can give you two answers, one for adding and one for subtracting You use numbers that can be really easy to do. 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. writing the number as a sum or difference you will have the same result, but during the equation the multiplication is a bit different.

7. Binomial expansion vs multiplication Binomial Expansion 99 2 = (100-1) 2 = 10000 – 2x100x1 + 1 = 9801 Faster and took smaller space Multiplication 99 2 = 8 99 x99 891 +8910 longer took space. 9801

8. Not always a good idea In some equations, our method becomes hard and messy, 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. 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. Sometimes there are decimals with numbers such as 1.3203 or 0.001410 which will really confuse you.

9. Examples 83.4 2 = (80+3+0.4) 2 = 80 2 +80x3+80x0.4+3x80+3 2 +3x0.4+0.4x80+0.4x3+ 0.4 2 = 80 2 +80x3+80x0.4+3x80+3 2 +3x0.4+0.4x80+0.4x3+ 0.4 2 = 6400+240+32+240+9+0.12 +32+0.12+0.16 = 6400+240+32+240+9+0.12 +32+0.12+0.16 = 6955.56 = 6955.56 83.4 x83.4 3336 25020 667200 6955.56

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

11. 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. 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.

12. Example 52 3 = (50+2) (50+2) (50+2) = 50 3 +50x2x2+2x50+2x50 +2x2x2 = 125000+200+100+100+ 8 = 140608 52 x52 104 2600 2704 x52 5408 135200 140608

13. 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

14. 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.