Binomial Expansion Reflection

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

2. Background Information • Our guiding or main questions was “is there an easy way to do 0.992?” We discovered we could do this through binomial expansion 0.992= (1-0.01) (1-0.01) = 12-2x1-0.01+(-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 = 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 • For the square of the difference of two numbers we developed: (a-b)2 = a2 - 2ab + b2 -This is slightly 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

4. Advantages 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? • An engineer 100 years ago could rely on this method as a shortcut. Since an engineer 100 years ago would not have had a calculator they would be using pencil and paper. • Shortcuts like this would therefore be important, as they increase efficiency and time productivity • This is especially important to someone like an engineer who is bound by a time schedule or a limit

5. To an Engineer…. • For someone like an engineer, this method is: • Useful, because it is very straightforward, and has many applications • Reliable, because it has little steps, with less room for error. For an engineer reliability is important • 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

6. Long Multiplication vs. Binomial Expansion (992) Binomial Expansion Long Multiplication 992= (100-1)2 = 10000 – 2x100x1 + 1 = 9801 ✔ 992= 899 x99 891 +8910 ✗ 9801

7. Explanation • This method lets us write the number as either the sum or difference of two numbers • The method allows you to multiply numbers you are comfortable with, in this case, 100 and 1. • Alternative to long multiplication, because the method is different, so it is shorter. • This is because writing the number as a sum or difference you end up with the same result, but during the process the multiplication is different

8. Further Explained… • 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 • If you are not using the algebraic method, you are still able to use this method, by following the distributive law

9. However… • In some situations, our method becomes cumbersome and messy such as: • When the numbers have many decimal places. This added to the whole numbers already to the left can get confusing • At the end, it is harder to know where to place the decimal point • Also, there are decimals which may have zeros, which are normally easily to multiply, but will later on give you more complicated decimals to add and multiply

10. Examples 96.02 x96.02 19204 5461200 86418000 9219.8404 96.022= (90+6+0.02)2 = 902+90x6+90x0.02+6x90+ 62+6x0.02+0.02x90+0.02 x6+0.022 = 8100+540+1.8+540+36 +0.12+1.8+0.12+0.04 = 9219.8404

11. Further Examples 83.4 x83.4 3336 25020 667200 6955.56 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

12. Other Situations • Other situations where using long multiplication is probably a better option than binomial expansion • Number to powers greater than two, i.e. cubed, to the power four, etc. • When numbers must be broken up into three ore more parts, such as three digit numbers • When multiplying three or more two-digit numbers • Numbers that have 3 or more digits

13. 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.

14. Examples 23 x21 23 460 483 x15 2415 4830 7245 • (23)(21)(15)= (20+3)(20+1)(10+5) = 202+202+20x10+20x1+20x5+3x20+3x1+3x10+3x5 = 400+400+200+20+100+60+3+30+15 =7245

15. Further Examples Pt. 2 523= (50+2) (50+2) (50+2) = 503+50x2x2+2x50+2x50+2x2x2 = 125000+200+100+100+8 = 140608 52 x52 104 2600 2704 x52 5408 135200 140608

16. Pascal’s T R I A N G L E • Is formed by adding adjacent numbers and writing the answer in the line under in a brick-fashion • First diagonal is formed from ones. Second is formed from counting numbers and the third is formed from triangular number. The white area is tetrahedral numbers • Horizontal Sums are equal to two to the power of the row. This is a very useful as an alternate method to finding large powers, instead of both long multiplication and binomial expansion

17. 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.