Binomial Expansions

1. Binomial Expansions By: Yeon

2. General Rule for Expanding Binomials: • Squaring the sum of 2 terms: (a+b) = a +2ab + b • Squaring the difference of 2 terms: (a – b) = a – 2ab + b

3. 100 years ago engineer • Since we did not have calculators 100 years ago, we had to look for simpler ways to do calculations. Calculators made everything easier for us, but back then they did everything with pencil and paper.

4. Binominal expansions Positives Binominal expansions could have been more useful in the olden days, rather than doing long division. It makes the equation easier and quicker to do, when using whole numbers. This is because: • It is a method that shortens the long maths equations • The method allows you to find the answer without going through every single multiplication equation. In the past, you had to do every single equation by hand, as a result it could get very messy, and get you confused. • When you find bigger numbers in situations, it is easier to use this method, it shortens the amount of work you do. • In addition, when finding the sum of the square, the number can be bigger than two different digits. It can be 561 not only 61.

5. Method is Cumbersome? – the negatives Personally, I found it a bit difficult when using this method for decimals, although that could be the same in long multiplication. Sometimes, it just won’t work in some situations. • When using the binominal method for whole numbers it is very easy, but for decimals it can get complicated, it works, but it confused me • It makes us think a lot, maybe even more compared to long multiplication, when applying the binominal method for decimals • When the equation contains 3 different numbers the rule becomes useless, for that question • When having two different signs, it cannot give the actual answer

6. Example 1 of where long multiplication is more efficient than the expansion method • Doing this is a faster method • (31) • = (30+1)(30+1) • = 900 + 30 + 30 + 1 (900 + 60 +1) • = 961 • Even if you had to add the line “30 x 30 + 30 x 1 + 1 x 30 + 1 x 1” the longest it could get would be: • (31) • = (30+1)(30+1) • = 30 x 30 + 30 x 1 + 1 x 30 + 1 x 1 • = 900 + 30 + 30 + 1 • = 961 • And this looks much neater and easier, to do • Rather than 31 x 31 = 31 x 31 1 x 1 = 1 1 x 30 = 30 30x 1 = 30 30 x 30 = 900 1 30 30 + 900 = 961

7. Example 2 • Doing this is a faster method • (27) • = (30-3)(30-3) • = 900 – 90 – 90 + 9 (900 – 180 +9) • = 729 • Even if you had to add the line “= 30 x 30 + 30 x (-3) + (-3) x 30 + (-3) x (-3)”, the longest it could get would be: • (27) • = (30-3)(30-3) • = 30 x 30 + 30 x (-3) + (-3) x 30 + (-3) x (-3) • = 900 – 90 – 90 + 9 • = 729 • Rather than 27 x 27 = 27 x 27 7 x 7 = 49 7 x 20 = 140 20 x 7 = 140 20 x 20 = 400 49 140 140 + 400 = 729

8. Example 5: Bigger Numbers • The Sum (201) = (200 + 1) (200 + 1) = 40,000 + 200 + 200 +1 = 40,401 And the longest it can get is: (201) = (200 + 1) (200 + 1) = 200 x 200 + 200 x 1 + 1 x 200 + 1 x 1 = 40,000 + 200 + 200 +1 = 40,401 • The Difference (299) = (300 – 1)(300 – 1) = 90,000 – 300 – 300 + 1 (90,000 – 600 + 1) = 89,401 And the longest it can get is: • (299) • = (300 – 1)(300 – 1) • = 300 x 300 + 300 x (-1) + (-1) x 300 + (-1) x (-1) • = 90,000 – 300 – 300 + 1 (90,000 – 600 + 1) • = 89,401

9. Example 7: Bigger Numbers (my extension) • Rather than 561 x 561 = 561 x 561 1 x 1 = 1 1 x 60 = 60 1 x 500 = 500 60 x 1 = 60 60 x 60 = 3,600 60 x 500 = 30,000 500 x 1 = 500 500 x 60 = 30,000 500 x 500 = 250,000 • Doing this is a cleaner method (561) • = (500 + 60 + 1)(500 + 60 + 1) = 250,000 + 30,000 + 500 + 30,000 + 3,600 + 60 + 500 + 60 +1 = 314,721 Even thought using a larger number, it doesn’t get so long and complicated, it’s clear and simple. Easier to understand and com back to. 1 60 500 60 3,600 30,000 500 30,000 + 250,000 314,721

10. Example 1 of where long multiplication is more efficient • One obvious on is that if the numbers are not the same, it cannot follow the same method. For example: (44 + 48)(43 + 67) = 1892 + 2948 + 2064 + 3216 = 10120 The actual answer 10120 is not equal to 5452, the answer got from using the binomial method. When having different numbers, if the binomial expansion method is applied, the numbers and answers go totally out of control, it is not even for one moment correct. Not: (44 + 48)(43+67) = 1036 + 2112 + 2304 a + 2ab + b = 5452 (I used the first 2 numbers, to prove my point.)

11. The long multiplication would be done like this: (44 + 48)(43 + 67) 44 x 43 1892 44 x 67 2948 48 x 43 2064 48 x 67 3216 The addition: 1892 2948 2064 + 3216 = 10120

12. Example 2: A continuous • Another Example would be: (22 + 41)(50 + 28) 22 x 50 1,100 22 x 28 616 41 x 50 2,050 41 x 28 1,148 • And another Example would be: (45 + 19)(39 + 19) 45 x 39 1,755 45 x 19 855 19 x 39 741 19 x19 361 1,100 616 2,050 + 1,148 = 4914 1,755 855 741 + 361 = 3,712

13. Example 3: Another • If the minus and addition signs don’t remain constant the binomial expansion cannot be used. For example: (21 + 34)(21 – 34) = 441 – 714 + 714 – 1156 = 441 – 1156 = -715 -715 is not at all equal to -1429, therefore long multiplication would be more efficient for these equations. The numbers in the middle cross each other out, making the rule untrue. So, with the sign change the binomial method cannot be applied, it is an invalid rule. But if you were to follow the rules it would be: (21 + 34)(21 – 34) = 441 – 714 – 1156 a – 2ab – b = -1429

14. The long multiplication would be done like this: Then the addition and subtraction: =441 – 714 + 714 – 1156 441 +714 = 1155 And: 1156 + 714 = 1870 So you can: 1155 -1870 = -715 First the multiplication: (21 + 34)(21 – 34) = 21 x 21 441 21 x -34 -714 34 x 21 714 34 x -34 • 1156 But, instead of all that from 441 – 714 + 714 – 1156, you could probably just take the negative and the positive 714’s out 

15. Example 4: Continuous • Another Example: (53 – 20)(83 + 29) 53 x 83 = 4,399 53 x 29 = 1,537 -20 x 83 = -1,660 -20 x 29 = -580 • Another Example: (94 + 18)(72 – 94) 94 x 72 = 6,768 94 x -94 = -8,836 18 x 72 = 1,296 18 x -94 = -1692 4,399 + 1,537 = 5,936 -1,660 + -580 = -2,240 5,936 – 2,240 = 3,696 6,768 + 1,296 = 8,064 -8,836 + -1,692 = -10,528 8,064 – 10,528 = -2,464

16. Conclusion: In conclusion, long multiplication and binomial expansions, both have their positives and negatives, in its own special ways. Long multiplication may be a bit easier to think of strait in your head (that may be what you think of), but when trying having numbers on a sheet, so for business, that you have to times itself by, to get the amount of money, etc, you would use binomial equations. Things have become much more convenient, because, now we have calculators, so we don’t have to do everything on paper or in our heads. Even if, perhaps I did not find the right answer for why long multiplication is more efficient, it should have a reason, otherwise there is no reason for us not to use binomial expansions, they are much easier and simpler.

17. Bibliography for images • http://www.amatteroffax.com/images/inventoryimages/1215883.jpg • http://webpages.uah.edu/~robinsc/pencil-n-paper.jpg • http://t1.gstatic.com/images?q=tbn:1acRU0Vo_pt-BM:http://www.bananagrams-game.com/animated-thumbs-up-happyfaces.gif&t=1 • http://school.discoveryeducation.com/clipart/images/stk-fgr2.gif • 2 I drew  • 3 from Clipart