Introducing
Your new presentation assistant.
Refine, enhance, and tailor your content, source relevant images, and edit visuals quicker than ever before.
Trending searches
By Abhinaya Uthayakumar, Julie Bu, and Emily Fong
A binomial is a polynomial with two terms. We're going to look at the Binomial Expansion Theorem, a shortcut method of raising a binomial to a power.
Do you see a pattern?
There are several things that you hopefully have noticed after looking at the expansion.
Pascal's Triangle, named after the French mathematician Blaise Pascal is an easy way to find the coefficients of the expansion.
Each element in the triangle is the sum of the two elements immediately above it.
Each row in the triangle begins and ends with 1.
A combination is an arrangement of objects, without repetition, and order not being important.
Another definition of combination is the number of such arrangements that are possible.
The n and r in the formula stand for the total number of objects to choose from and the number of objects in the arrangement, respectively.
Let's consider the n=4 row of the triangle.
4C0 = 1, 4C1 = 4, 4C2 = 6, 4C3 = 4, 4C4 = 1
Notice that the 3rd term is the term with the r=2. That is, we begin counting with 0.
The Binomial Expansion Theorem can be written in summation notation, where it is very compact and manageable.
Remember that since the lower limit of the summation begins with 0, the 7th term of the sequence is actually the term when k=6.
The x starts off to the nth power and goes down by one each time, the y starts off to the 0th power (not there) and increases by one each time. The coefficients are combinations.
Expand ( 3x - 2y )^5:
Start off by figuring out the coefficients. Remember that these are combinations of 5 things, k at a time, where k is either the power on the x or the power on the y (combinations are symmetric, so it doesn't matter).
C(5,0) = 1; C(5,1) = 5; C(5,2) = 10; C(5,3) = 10; C(5,4) = 5; C(5,5) = 1
Now throw in the 3x and -2y terms.
Raise the individual factors to their proper powers.
Simplify each term to get the final answer.
MLA Bibliography
"Exponents." Test Page for Apache Installation. Web. 05 May 2011. <http://hyperphysics.phy-astr.gsu.edu/hbase/alg3.html>.
Binomial Theorem: The algebraic expansion of powers of a binomial
The coefficient, a, can be determined by refering to Pascal's triangle.
a = coefficient
b = nonnegative integer
c = nonnegative integer
b + c = n
Do you see a pattern?
For example,
(x+y)^5 = x^5 + 5x^(4)y + 10x^(3)y^(2) + 10x^(2)y^(3) + 5xy^4 + y^5
There are many patterns involved with the binomial expansions. Some include:
The binomial expansion theorem can be represented by:
A binomial is a polynomial with two terms. We're going to look at the Binomial Expansion Theorem, a shortcut method of raising a binomial to a power.
Powers such as (x + y)^n are able to be expanded into a sum of terms in the format ax^(b)y^(c).
Pascal's Triangle, named after the French mathematician Blaise Pascal, is an easy way to find the coefficients of the expansion.
Each row in the triangle begins and ends with 1. Each element in the triangle is the sum of the two elements immediately above it.
Remember combinations and permutations? Well, combinations help us with the Binomial Expansion Theorem.