Exploring the Binomial Expansion Theorem Introduction In algebra binomial expansion is the expansion of powers of a binomial. A binomial expansion is an expression in which it contains two terms eg, (a+b). This expression could also have a power on the outside of the brackets. Aim To generate a formula for finding the general expanded form of binomial expressions of the form (a+b)n. (Source The Sheet) Basic Binomial Expansions (a+b)1 = a+b (a+b)2 = a2+2ab+ b2 (a+b)3 = a3+ 3a2b + 3ab2 + b3 (a+b)4 = a4+ 4a3b + 6a2b2+ 4ab3+ b4 The power (n) and the number of terms in each expansion is equal to the amount of terms in each expansion plus one. The coefficients in each binomial expansion is a line on Pascal’s triangle. The nth power is the number line the coefficients are on. (a+b)1 = (1)a+b (a+b)2 = (1)a2 + (2)ab + (1)b2 (a+b)3 = (1)a3+ (3)a2b + (3)ab2 + (1)b3 (a+b)4 = (1)a4+ (4)a3b + (6)a2b2+ (4)ab3+ (1)b4 If one was to predict the coefficients in (a+b)7 it would be the seventh line on Pascal’s triangle. The coefficients are 1, 7, 21, 35, 35, 21, 7, 1. The indices on a and b both have their pattern. Notice how the indices for a on (a+b)4 go 4,3,2,1,0 and the indices on b go 0,1,2,3,4. This pattern can be seen in any (a+b)n form. The n in the expression represents what the power for a and b would start and go down to or vice versa. If one was to fully expand (a+b)15 and to find all coefficients, nCr needs to be used (where n is the indice on the equation, and r is a number between 0 and n). If one was to do this manually, the line the equation lies on is the nth term (in this case it lies on the 15th line) The first expression using nCr would be 15C0 which equals 1. Further expansion using nCr is shown below. ... ... middle of paper ... ...iven equation. A pattern in the indices was also found, this added to the efficiency of the expansion. If all these factors are brought together, a formula has been discovered that finds the general expanded form of . To further this investigation, one could look into a formula that works for negative indices and compare this to positive integer formula. A pattern could be found from the two results and a formula could be created that works and generates answers for both positive and negative indices. Works Cited https://www.youtube.com/watch?v=ktDy4ExrctQ https://www.youtube.com/user/ExamSolutions?feature=watch http://www.wolframalpha.com/widget/widgetPopup.jsp?p=v&id=6767c31553b5f2c292b426c97a9b6013&title=Binomial%20Expansion%20Calculator&theme=blue&i0=a%2B2b%2B3x&i1=6&podSelect=&includepodid=Input&includepodid=Result http://ptri1.tripod.com/
On the second day of class, the Professor Judit Kerekes developed a short chart of the Xmania system and briefly explained how students would experience a number problem. Professor Kerekes invented letters to name the quantities such as “A” for one box, “B” for two boxes. “C” is for three boxes, “D” is for four boxes and “E” is for five boxes. This chart confused me because I wasn’t too familiar with this system. One thing that generated a lot of excitement for me was when she used huge foam blocks shaped as dice. A student threw two blocks across the room and identified the symbol “0”, “A”, “B”, “C”, “D”, and “E.” To everyone’s amazement, we had fun practicing the Xmania system and learned as each table took turns trying to work out problems.
An exponential equation is a type of transcendental equation, or equation that can be solved for one factor in terms of another. An exponential function f with base a is denoted by f (x) = ax, where a is greater than 0, a can not equal 1, and x is any real number. The base 1 is excluded because 1 to any power yields 1. For example, 1 to the fourth power is 1×1×1×1, which equals 1. That is a constant function which is not exponential, so 1 is not allowed to be the base of an exponential equation. Otherwise, the base of a can be any number that is greater than 0 and isn’t 1, and x can be any real number. The equation for the parent function of an exponential functions follows as so:
Cu (aq) + 2NO3 (aq) + 2Na+ (aq) + 2OH- (aq) → Cu(OH)2 (s) + 2Na+ (aq) + 2NO3(aq)
"magic number" 7 plus or minus 2 - that is between 5 and 9 bits of
The order of operations works like this: First anything in the parentheses, then we do the exponents/roots, then any multiplication and division- which is done in that order, then we do Addition and Subtraction- in that order as well. To explain this, we will solve the problem above: Step 1. The first thing you do in the order of operations is to do anything listed in parentheses, but you must also keep in mind everything else. So the first set we do is (5+2), even though it is the last set, addition comes first on the order of operation list. So, (5+2)= 7 right?
Kimberling, William C. “The Electoral College.” Federal Election Commission, May 1992. Web. 13 March 2012.
So now that we have the desired answer and know what the formula is, we just need to know what it means and why it works. So
x 3, 4 x 4 x 4, 5 x 5 x 5, 6 x 6 x 6, 7 x 7 x 7, 8 x 8 x 8, 9 x 9 x 9)
as the “r-value” and “r” can be any value between -1 and +1. It can be
The Electoral College is a confusing topic to most people, and its effect on how votes are represented in presidential elections. Essentially people vote and electors, people assigned to vote on the people 's behalf, and the candidate that wins the popular election in that state 's gets the the Electoral College votes for that state. The amount of votes is based on the population and the first candidate to win two hundred and seventy Electoral College votes wins. This system is debated on whether it benefits or hinders the election process, and how it does this is also debated by political experts.
... the new possibility for 3 things going 3 at a time: “ab” combined with “c” becomes “abc.”
it is a result of a vigorous study of each of the components of the equation of
The Fibonacci sequence was introduced as a problem involving population growth based on assumptions. Fibonacci got the idea from early Indian and Arabian Mathematics. He grew the theory and introduced it to the western world. The sequence is explained by starting at 1, 2 then adding the two t...
in exponential form. For instance, in a base 2 system, 4 can be written as 2
One only requires the use of simple mathematics i.e. simple addition and subtraction of single and double digit numbers ...