Solving Binomial Expansion

In college algebra is the part of arithmetic, which perform the calculation on the variables. In the algebra, expressions are written in the form polynomial which is a part of secondary board of education andhra pradesh. The binomial polynomial (polynomials worksheet) contains two terms. These two terms can be considered as a collection of or arranging the two monomial with operators and brackets option. Through this session we are discussing about Solving Binomial Expansion. Dividing Polynomials can be defined as two variables which perform some expression or some operation. Like a² – b² = (a + b) (a – b)
The algebra provides the lots of way to solve the binomial expression. In the mathematics to solve the binomial expression we use the concept of expansion of binomial theorem. Binomial theorem can generally be represented by the formula given by Blaise Pascal in the 17th century. The most generic example of binomial expression is given below:
(a + b)² = a² + 2ab + b²
 In the more general term binomial formula can be expressed as a:
     (a + b)ⁿ = ∑∞i=0  (n/i) aⁱ b^n-i
Here (n/i)  is a binomial coefficient and ‘n’ is a real number.
In the simple term binomial expression can be written as:
(a + b)² = a² + 2ab + b²
(a + b)³ = a³ + 3a²b + 3ab²  + b³
As same like binomial theorem formula can be written as:
(a + b)ⁿ = aⁿ + n(a^n-1 b¹) + n(n-1)/2! (a^n-2 b²) + n(n-1) (n-2)/3! (a^n-3 b³)+……….+ bⁿ

Here we will show you binomial expansion examples and how to solve the binomial expression. (Know more about Binomial Expansion in broad manner, here,)
Example1: Solve the binomial given expression (a + 5)³.
Solution:  Given that (a + 5)³, as we can see that it’s a binomial expression. So this can be solved by following the binomial expansion formula:
So, solution become in simplified formula:
      (a + 5)³ = a⁵ + 3x² (5) + 3x (5)² + 5³  
      (a + 5)³ = a⁵ + 15 x² + 75x + 125
As clear from the above solution, we can easily solve the binomial related problems.
In the next session we are going to discuss Binomial Experiment and Read more maths topics of different grades such as  How to do Estimate Quotients in the upcoming sessions here.

Binomial Expansion

Hello students ,Previously we have discussed about column multiplication and in this blog we are going to discuss about the Binomial Expansion which is a part of secondary school board andhra pradesh. Binomial Expansion is related to the algebraic expansion of powers of a Binomial Distributions. We have a case when n is a positive integer then the expansion of ( 1 + a )ⁿ is equal to ∑ⁿ_r=0 c_rⁿ a^r. Coefficients of 'a' that appear in the expansion of ( 1 + a )ⁿ are known as binomial coefficients. (Know more about Binomial Expansion in broad manner, here,)

By using the formula of expansion you can easily expand the series without doing the multiplication. There are mainly two properties of binomial expansion that include :

1. ( n + 1 ) terms contained by an expansion .
2. Binomial coefficients c_rⁿ should be integers .
   We can understand it by an example as ( a + b )² is described in terms of expansion as
   a² + 2ab + b² where ab is the coefficient .
   Binomial Expansion examples :
   ( a + b ) ⁴ : It expands as a⁴ + 4 a³ b + 6 a² b² + 4 a b³ + b⁴ . Here in each term exponents of a and b are non negative integers with sum of the powers of a and b is equal to n. In above example n is equal to the 4 .
   Binomial Expansion can be understood using Pascale's triangle that applies to expand the terms in form of ( a + b ) ⁿ .
                                                                        1

                                                                         1  1
                                                                        1 2 1
                                                                       1 3 3 1
                                                                      1 4 6 4 1
                                                                    1 5 10 10 5 1
                                                                  1 6 15 20 15 6 1
We can take an example to understand the Pascale's triangle as ( a + b )³ is expanded by taking the row of triangles starting with 1 and 3 is 1 3 3 1 therefore expansion of ( a + b )³ is described as
( a + b )³ = a³ + 3 a²b + 3 ab² + b³ .
In the next session we are going to discuss Solving Binomial Expansion
and if anyone want to know about Math Blog on Estimating Quotients then they can refer to Internet and text books for understanding it more precisely.