The binomial theorem is a simplified way of finding the expansion of a binomial to a certain power. We can of course find the expanded form of any binomial to a certain power by writing it and doing each step, but this process can be very time consuming when you get into let’s say a binomial to the 10th power. Example:

(x+y)^0=1 of course because anything to the power if 0 equal 1 (x+y)^1= x+y anything to a power of 1 is just itself.
(x+y)^2= (x+y)(x+y) NOT x^2+y^2.
So expand (x+y)(x+y)=x^2+xy+yx+y^2 or x^2+2xy+y^2.
(x+y)^3=(x+y)(x+y)(x+y) now expanding it is getting quite long. Of course we could do this using the distribution property, but there must be an easier way of expanding binomials with out doing all the steps that is takes to expand something like (x+y)^10.

As surprising as it might be, there is an easier way of find what a binomial equals to a larger power. Using combinations we can find the coefficients of each term. Lets look at an example. The (x+y)^3 was the one I didn’t finish. Let’s look at it now. Using the equation in combination, we can insert the power that we are using and for each term to find the coefficient of each term. Ex:

This process can be even easier. Blaise Pascal, a famous French mathematical among other things put together a triangle made up of numbers where each number represents the coefficient of each term (when expanded) in a binomial to a certain power. He named the triangle “triangle du arithmétique” or in English, “The Arithmetical Triangle”. Now the triangle is called “Pascal’s triangle named after the creator, Blaise Pascal. We can find this triangle by using combinations. Ex:

Moving back to the binomial theorem, you can use any binomial and find what it equals using Pascal’s triangle or combinations. We can us any two random numbers, a number and a variable, variables with coefficients, subtraction instead of adding etc. Lets look at some examples.

...CHAP 1 - Binomial Expansions (Kembangan Binomial)
The binomialtheorem describes the algebraic expansion of powers of a binomial.
Figure 1 : Example use the binomial Expansion in geometric
There are 3 methods to expand binomial expression
Method 1 - Algebra method
Expansion two or more expression.
Example: The expansion depend on power value (n)
n = 0, (a +...

...BINOMIALTHEOREM
OBJECTIVES
Recognize patterns in binomial expansions.
Evaluate a binomial coefficient.
Expand a binomial raised to a power.
Find a particular term in a binomial expansion
Understand the principle of mathematical induction.
Prove statements using mathematical induction.
Definition: BINOMIALTHEOREM
Patterns in Binomial Expansions...

...1 10/10/01
Fermat’s Little Theorem From the Multinomial Theorem
Thomas J. Osler (osler@rowan.edu) Rowan University, Glassboro, NJ 08028 Fermat’s Little Theorem [1] states that n p −1 − 1 is divisible by p whenever p is prime and n is an integer not divisible by p. This theorem is used in many of the simpler tests for primality. The so-called multinomial theorem (described in [2]) gives the expansion of a multinomial to...

...this as the sample size in the Binomial Probability Distribution feature of PhStat.
The probability of the event, that a drive would fall below Four-D’s quality standard of 6.2, was gained from Question 1.
Outcomes 1 – 10 were queried because there are 10 possible scenarios of Four-D rejecting the sample. The cumulative probability of Four-D rejecting a shipment from DataStor’s “in control” process is 3.8%. See the table below for the calculation.
The...

...#1 True or false: Even if the sample size is more than 1000, we cannot always use the normal approximation to binomial.
Solution:
If a sample is n>30, we can say that sample size is sufficiently large to assume normal approximation to binomial curve.
Hence the statement is false.
#2
A salesperson goes door-to-door in a residential area to demonstrate the use of a new Household appliance to potential customers. She has found from her years of...

...The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students.
The Pythagorean Theorem is...

...-------------------------------------------------
Pythagorean Theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two...

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