26 ## MULTIPLICATION OF SUMS## A proof of the binomial theoremTHE BINOMIAL THEOREM gives the coefficients in the product of ( If we actually multiplied the 4 factors of ( then before adding the like terms, we would find terms in
The binomial theorem tells ( The final product will be a polynomial of the 3rd degree:
As with the binomial theorem, the question is: What are the coefficients? They will be some function of the constants 1, 2, 3. For the moment, we will simply show the result:
The coefficient of The coefficient of And the constant term is their combination taken Why those combinations? We will see why as we continue. For the moment, the student should attempt this problem.
Problem 1. Multiply out ( To see the answer, pass your mouse over the colored area.
Let us now begin again, and analyze the multiplication of these elementary sums: ( Upon multiplying, we would find six terms. Each term will contain two factors, namely one letter from each factor:
Therefore, we can write the product of the following -- ( -- simply by writing the sum of all combinations of one letter from each factor.
Each term in the product consists of three factors: one from each binomial. Note that there are a total of 2 Multiplication of n binomials produces 2 For, multiplication of two binomials gives 4 terms: ( If we multiply those with a binomial, we will have 8 terms; those multiplied with a binomial will produce 16 terms; and so on. Now consider these four binomial factors, in which each of the constants, ( And let us compare it with ( in which each constant is the same. In this product -- ( -- each of the 2
and so on. That is, we will find terms in Now, how is a term Similarly, then, in the expansion of ( the coefficient of ( Next, terms with
The coefficient of
In the expansion of ( then, terms with
The coefficient of ( Next, terms with ( As for terms in ( the only difference is that all the constants are the same. There will be ( The coefficient of ( A term in ( There will be In the expansion of ( there will similarly be ( The coefficient of ( Finally, the constant term will be produced by taking the letter from each of the 4 factors. There is
As for the constant term in the expansion of ( again there will be
We have found: (
We have in this way accounted for the binomial coefficients in the epansion of ( The binomial coefficients are how many terms there are of each kind. The result is general. The binomial theorem states that in the expansion of
(
Problem 2. Imagine multiplying out ( a) How many terms would there be?
3 b) Each term would consist of how many factors? Two
Problem 3. Imagine multiplying out ( a) How many terms would there be?
2 Thus a product of b) Each term would consist of how many factors? Four c) How is a term produced that contains three factors of
By taking d) Therefore, what is the coefficient of e) What is the coefficient of f) What is the coefficient of g) What is the coefficient of h) What is the constant term?
Problem 4. Multiply out by taking the correct combinations of the integers. a) ( b) (
Problem 5. In this multiplication (
Problem 6. ( a) Upon multiplying out, and before collecting like terms, how many
b) How will a term
By taking c) How many
Problem 7. In each row of Pascal's triangle, the sum of the binomial coefficients is 2
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