1.  Binomial Expansion Reflection Hossam Khattab, Grade 8B Qatar Academy November 3 rd, 2010

2. Background Information  Our guiding or main questions was “is there an easy way to do 0.99 2 ?” We discovered we could do this through binomial expansion 0.99 2 = (1-0.01) (1-0.01) = 1 2 -2x1-0.01+(-0.01) (-0.01) = 0.9801 This method is much quicker, and less hassled than using long multiplication

3. General Rules  For the square of the sum of two number we developed: (a+b) 2 = a 2 + 2ab + b 2  For the square of the difference of two numbers we developed: (a-b) 2 = a 2 - 2ab + b 2

4. Usefulness as Opposed to Traditional Multiplication  If you were an engineer 100 years ago, explain how our method may have been useful rather than just long multiplication?  Using this method is much quicker, especially for numbers that are close to tens, hundreds, etc.  The number allows you to have another way to check your answer, especially since this method is highly reliable

5. Long Multiplication vs. Binomial Expansion Binomial Expansion 99 2 = (100-1) 2 = 10000 – 2x100x1 + 1 = 9801 ✔ Long Multiplication 99 2 = 8 99 x99 891 +8910 ✗ 9801

6. Explanation  In the case shown previously, binomial expansion is shorter and easier, because the number is very close to a hundred  The method is useful because is takes less time, and is easier to do therefore less prone to error  It helps spread the numbers out in a way that makes multiplication very simple, because you are multiplying numbers that may have lots of digits, but the most of these digits are zeros

7. However…  In some situations, our method becomes cumbersome such as: -> In situations with numbers that have many decimal places, where the amount of zeros involved becomes a problem -> In situations where the number has many digits, and those digits are not zeros (digits including those after a decimal point) -> In situations where it is not squaring, but it is just multiplying two large numbers

8. Examples (8976)(8867)= (8000+900+70+6)(8000+800+60+7)= 8000 2 +8000x600+8000x60+8000x7+ 900x8000+900x800+900x60+900x7+ 70x8000+70x800+70x60+70x7+6x 8000+6x800+6x60+6x7= 79590192 (96.020405) 2 =(100- 4+0.02+0.0004+0.000005) 2 = 100 2 +100(-4)+100x0.02+100x 0.0004+100x0.000005+100(-4)+(- 4) 2 +(-4)x0.02+(-4)x0.0004+(- 4)x0.000005 +0.02x100+0.02(- 4)+0.02 2 +0.02x 0.0004+0.02X0.000005+0.0004x100 +0.0004(-4)+0.0004x0.02+0.0004 2 + 0.0004x0.000005+0.000005x100+0.00 0005x0.02+0.000005x0.0004+ 0.000005 2 = 9219.918176364025

9. Other Situations  Other situations where binomial expansion is not useful, and where long multiplications is definitely the way to go: - Numbers that are not squared, but cubed, or powers larger than that - Situations where the number has to be broken up into more than a+b (more than two numbers) - When multiplying three or more two-digit numbers

10. Limitations Explained  When numbers start getting into 4, or even just 3 digits, this method becomes hard, and defeats the purpose of the mental math, because it will involve complex additions, and multiplications, and you will probably end up using long multiplication to calculate within the original calculation  When the number has to be split into more parts, the algebraic rule cannot work. Therefore you must use regular multiplication and expansion. You end up in turn, having to multiply every number by every number. This increases greatly every time you add a single digit to either of the two numbers involved.

11. Examples  52 3 = (50+2) (50+2) (50+2) = 50 3 +50x2x2+2x50+2x50+2x2 x2 = 140608  (23)(21)(15)= (20+3)(20+1)(10+5) = 20 2 +20 2 +20x10+20x1+20x5+3x2 0+3x1+3x10+3x5 =7245  233 2 =(200+30+3) (200+30+3) = 200 2 +200x30+200x3+30x200 +30 2 +30x3+3x200+3x30+3 2 = 54289

12. Conclusion Binomial Expansion is very useful in general for multiplying 2 or 3 digit numbers, and squaring them, which would usually be difficult or would require long multiplication It is a shortcut method, to reduce working for products where long multiplication would otherwise be necessary, however it cannot completely replace long multiplication, simply because it starts to get confusing with many decimals, large numbers, and larger powers.