Essay about Exploring Binomial Expansion Theorem

Essay about Exploring Binomial Expansion Theorem

Length: 1023 words (2.9 double-spaced pages)

Rating: Strong Essays

Open Document

Essay Preview

Exploring the Binomial Expansion Theorem


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.

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

Need Writing Help?

Get feedback on grammar, clarity, concision and logic instantly.

Check your paper »

Mathematical Exploration: Exploring the Proofs of Fermat's Little Theorem

- Unraveling the complex and diverse nature of numbers has always been a fascinating ordeal for me; that is what makes and keeps me interested in the world of mathematics. Finding out new number patterns and the relationship between numbers is nothing short of a new discovery; that is how my interest into learning more about and exploring Fermat's Little Theorem came about. INTRODUCTION OF FERMAT'S LITTLE THEOREM Pierre de Fermat was a French mathematician whose contribution to analytic geometry and calculus are duly noted....   [tags: little theorem, binomial theorem, pierre de fermat]

Strong Essays
1381 words (3.9 pages)

Essay The 's Last Theorem, By Michael Hutchings

- Fermat’s Last Theorem--which states that an + bn = cn is untrue for any circumstance in which a, b, c are not three positive integers and n is an integer greater than two—has long resided with the collection of other seemingly impossible proofs. Such a characterization seems distant and ill-informed, seeing as today’s smartphones and gadgets have far surpassed the computing capabilities of even the most powerful computers some decades ago. This renaissance of technology has not, however, eased this process by any means....   [tags: Mathematics, Logic, Theorem, Scientific method]

Strong Essays
1742 words (5 pages)

Mathematics: Pascal's Triangle Essay

- Pascal’s Triangle is a visual represenation a series of binomial expansions. The triangle emerges as a result of the function (x + y) ^n where n is an integer greater than or equal to zero. As n increases, the quantity of terms in the result increases: 1. (x + y)^0 = 1………………………………………………………………………………. one term 2. (x + y)^1 = x + y………………………………………………………………………… two terms 3. (x + y)^2 = x^2 + 2xy + y2……………………………………………………………. .three terms Additionally, the integers represented on the triangle are found as the coefficients of the expansion....   [tags: binomial expansions, integer]

Strong Essays
957 words (2.7 pages)

Fermat’s Little Theorem Essay

- 1. Introduction: As I was looking for a theorem to prove for my Mathematics SL internal assessment, I couldn’t help but read about Fermat’s Little Theorem, a theorem I never heard of before. Looking into the theorem and reading about it made me develop an interest and genuine curiosity for this theorem. It was set forth in the 16th century by a French lawyer and amateur mathematician named Pierre de Fermat who is given credit for early developments that led to infinitesimal calculus. He made significant contributions to analytic geometry, probability, and optics....   [tags: prime number, mathematics]

Strong Essays
864 words (2.5 pages)

Analysis of Global Expansion and Interaction by David Ringrose Essay

- Global Expansion and Interaction by, David Ringrose is about global history and civilization in Latin America, Africa, and Asia. World expansion between 1200 and 1700 offers a helpful perspective on the world since 1950. (Pg.4) The theme presented in the book is the global history during 1200 to 1700 and is displayed in a cross-cultural and comparative manner. By examining the five key fields of conflict, from Imperial China to the Aztec and Inca Empires, he demonstrates how cultural, economic, and political areas of impact overlapped and expanded....   [tags: history, civilization, expansion]

Strong Essays
669 words (1.9 pages)

Essay on Kate Camp 's Poem Unfinished Love Theorem

- The online Oxford Dictionary states that a theorem is a “general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths” (OUP). Kate Camp’s poem Unfinished Love Theorem explores the various facets of love, in order to formulate a theory about it. The poem takes the reader on a journey through these discrete theories in an attempt to prove what love is. That it is an ‘unfinished’ theory suggests love is a complex emotion and therefore hard to define....   [tags: Love, Stanza, Present tense, Metaphor]

Strong Essays
1064 words (3 pages)

Essay Analysis Of Coase Theorem And Hardin 's Tragedy Of The Commons

- All of the case studies presented show a unique mixture of issues stemming from property rights, public goods, externalities, interjurisdictional spillovers and a fantastic illustration of Coase Theorem and Hardin’s Tragedy of the Commons. Water usage and rights are a pertinent and urgently growing issue that often pits economic development, sustenance and environmental health externalities at odds with each other. Water is needed to sustain life and ecosystems and property/jurisdictional rights regarding bodies of water are hard to distinguish....   [tags: Public good, Externality, Market failure]

Strong Essays
1234 words (3.5 pages)

Herbrandss Theorem Essay

- Herbrand’s Theorem Automated theorem proving has two goals: (1) to prove theorems and (2) to do it automatically. Fully automated theorem provers for first-order logic have been developed, starting in the 1960’s, but as theorems get more complicated, the time that theorem provers spend tends to grow exponentially. As a result, no really interesting theorems of mathematics can be proved this way- the human life span is not long enough. Therefore a major problem is to prove interesting theorems and the solution is to give the theorem provers heuristics, rules of thumb for knowledge and wisdom....   [tags: essays research papers]

Free Essays
1808 words (5.2 pages)

Bayes' Theorem Essay

- Bayes' Theorem I first became interested in Bayes' Theorem after reading Blind Man's Bluff, Sontag (1998). The book made mention how Bayes' Theorem was used to locate a missing thermonuclear bomb in Spain in 1966. Furthermore, it was again used by the military to locate the missing submarine USS Scorpion (Sontag, pg. 97) that had imploded when it sank several years later. I was intrigued by the nature of the theory and wanted to know more about it. When I was reading our textbook for the class, I came across Bayes' Theorem again, and found an avenue to do more research....   [tags: Papers]

Strong Essays
3823 words (10.9 pages)

Essay on Pythagoras' Theorem

- Pythagoras' Theorem I am going to study Pythagoras' theorem. Pythagoras Theorem is a2 + b2 = c2. 'a' being the shortest side, 'b' being the middle side and 'c' being the longest side (hypotenuse) of a right angled triangle. For example, I will use 32 x 42 = 52 . This is because: 32 = 3 x 3 = 9 42 = 4 x 4 = 16 52 = 5 x 5 = 25 So.. 9 +16 = 25 For this table, I am using the term a, b, b + 1 Triangle Number (n) Length of shortest side Length of middle side Length of longest side Perimeter Area 1 3 4 5 12 6 2 5 12 13 30 30 3 7 24 25 56 84 4 9 40 41 90...   [tags: Papers]

Free Essays
675 words (1.9 pages)