1. By: Juan Fernando Polanco 8A

2. BINOMIAL EXPRESSIONS  In algebra, we use letters to replace numbers  This allows us to apply the equations to different circumstances, instead of only one  A binomial is a polynomial expression with two terms like “a” and “b” or “x” and “y”  A binomial expression, along with a potency, may be very complex and so we use “Binomial Expansions” to make it easier to handle

3. BINOMIAL EXPANSION  This means expanding the binomial expression by writing all the terms within a potency  We get a different but equivalent term, meaning that it is the same thing but written in different words  This new term is easier to handle than the original one in certain cases  And now we can work more easily with the new one

4. BINOMIAL EXPANSION  (a+b) 2 =  (a+b).(a+b) =  a(a+b)+ b(a+b) =  a 2 +ab+ba+b 2 =  a 2 +2ab+b 2  This is an example of a binomial expansion step by step  We use ingenuity to find different ways to solve a problem

5. THE OLD TIMES  Early mathematicians did not have calculators  Some multiplications were too complex to handle  They came up with smart ideas to cope with this, just as the binomial expansions  This reduced the complexity of the calculations and allowed them to work in real life

6. IS IT USEFUL?  Depending on the individual calculation, we use the most appropriate method  With simple numbers we may use the original expression (Long multiplication): (3+2) 2 =5 2 =5.5=25  With more complex cases we may use the binomial expansion  We may start from creating the binomial expression and then expand it to solve our problem

7. IS IT USEFUL?  0.99 2  =(1-0.01) 2  =1 2 -2(1 x 0.01)+0.01 2  =1-0.02+0.0001  =0.9801  This makes the calculation easier when we do not have a calculator

8. IT MAY NOT BE APPROPRIATE  On other cases it may be worse  Let’s compare the two ways to solve the same problem  In the first place we are going to use the binomial expansion  Let’s say that the original binomial is (2+2a) 2

9. SOLVE BY EXPANDING  (2+2a) 2  =2(2+2a)+2a(2+2a)  =4+4a+4+4a 2  =8+  This makes it a longer and more complicated way to solve this problem

10. SOLVING USING LONG MULTIPLICATION  (2+2) 2  =4 2  =4x4  =8  In this case it is easier to solve it directly  Binomial expansion is only best for small numbers but long equations  Long multiplication is best for big numbers, small equation.

11. BENEFITS  Binomial Expansion: The benefits of using binomial expansion are that you keep your work in a more organized way, and it’s easier to deal with small number but big equations: (4a+6+9b+b+ab+7) 2 +(b+4ª+5+8b+9+10)  Long multiplication: The benefits if using long multiplication are that your results are more accurate, and when it comes to big numbers but small equations it is more organized: x 12345678 87654321

12. LIMITATIONS  Binomial Expansion: Binomial expansion is organized but your results might not be as accurate for multiplications have to be done in your head, expanding the problem just helps you to make it easier to resolve since its more organized  It is also very hard to operate with really big and really small numbers is just ruins the whole process: (123456789+987654321) 2  Long multiplication: Long multiplication is not very organized with lots of numbers, so it is much easier to use large numbers but small equations, your results are much more accurate: 185 x 12 x 32 x 48 x 64 x 49 x 83 x 6

13. PASCAL’S TRIANGLE  So we found out how both solving process are better for different occasions, and we talked about binomial expansions 100 years ago, but even using binomial expansion for an equation with a high potency such as 4 and up becomes endless to solve  Equations with potency of 2 and 3, are just right for binomial expansion

14. PASCAL’S TRIANGLE (a+b) 3 =a (a+b)+b (a+b) =(a 2 +ab+ba+b 2 )(a+b) =a (a 2 +ab+ba+b 2 )+b (a 2 +ab+ba+b 2 ) =a 3 +a 2 b+ba 2 +ab 2 +ba 2 +ab 2 +b 2 a+b 3 =a 3 +3a 2 b+3ab 2 +b 3 So as you can see this was only to the power of 3 and look how complicated it becomes, so a french mathematician called Blaise Pascal made and easier method to solve high potency equations, such as 4, 6, 10, etc.

15. PASCAL’S TRIANGLE So what this numbers mean, is the coefficients for each potency, so its showing how many variables go and in what order and it becomes pretty easy to answer after some point where the potency is just humongous. Notice how this numbers are the coefficients for (a+b) 2 =a 2 +2ab+b 2 And obviously as you go down the potency becomes bigger and bigger. (a+b) 3 = 1= a 3 + 3= 3a 2 b + 3= 3ab 2 + b 3 =a 3 +3a 2 b +3ab 2 +b 3

16. BIBLIOGRAPHY  http://en.wikipedia.org/wiki/Pascal%27s_triangl e#Binomial_expansions http://en.wikipedia.org/wiki/Pascal%27s_triangl e#Binomial_expansions  http://www.youtube.com/watch?v=- fFWWt1m9k0&feature=channel http://www.youtube.com/watch?v=- fFWWt1m9k0&feature=channel  http://www.youtube.com/watch?v=Cv4YhIMfbe M&feature=channel http://www.youtube.com/watch?v=Cv4YhIMfbe M&feature=channel  http://www.google.com/images?um=1&hl=en& safe=active&tbs=isch:1&&sa=X&ei=5FnRTIW 1MIOougOsufnGDA&ved=0CCQQvwUoAQ&q =pascal%27s+triangle+wikipedia&spell=1&biw =1280&bih=560&safe=active http://www.google.com/images?um=1&hl=en& safe=active&tbs=isch:1&&sa=X&ei=5FnRTIW 1MIOougOsufnGDA&ved=0CCQQvwUoAQ&q =pascal%27s+triangle+wikipedia&spell=1&biw =1280&bih=560&safe=active