25 ## THE BINOMIAL THEOREMFormal statement of the theorem THE BINOMIAL THEOREM shows how to calculate a power of a binomial -- ( For example, if we actually multiplied out the 4th power of ( ( -- then on collecting like terms we would find: (
The degree of each term is 4. The first term is actually Thus to "expand" ( ( The question is, What are the coefficients? They are called the binomial coefficients. In the expansion of The answer to the question, "What are the binomial coefficients?" is called the binomial theorem. It shows how to calculate the coefficients in the expansion of ( The symbol for a binomial coefficient is . The upper index The lower index For example, when
( Again, each lower index is the exponent of Now, what are these binomial coefficients, ? The theorem states that the binomial coefficients are none other than the combinatorial numbers, = The binomial coefficients here are 1 5 10 10 5 1. Note the symmetry. The coefficient of the first term is always 1, and the coefficient of the second term is the same as the exponent of ( Note also that, when the combinatorial number is written as factors, each coefficient contains the previous one. The coefficient of That is, it is multiplied by the exponent of the previous The coefficient of Using sigma notation and factorials for the combinatorial numbers, here is the binomial theorem: What follows the summation sign is the general term. Each term in the sum will look like that -- the first term having Notice that the sum of the exponents ( Example 1. a) The term
b) In that expansion, what number is the coefficient of
Note again: The lower index, in this case 4, is the exponent of This same number is also the coefficient of
Example 2. Expand ( ( For,
When Each odd power of
Example 3. In the expansion of (
The coefficient of
Example 4. Write the first four terms of (
Notice: Each coefficient is a factor of the next coefficient. The coefficient . That in turn is a factor of . To construct the next coefficient, then, multiply the present coefficient by the exponent of
-- namely
And divide it by 1 more than the exponent of That is the coefficient of
Example 5. Use the binomial theorem to expand ( ( The first coefficient is always 1. The second is always the exponent of the expansion, in this case 8. The next coefficient can be constructed as described above. It will be the present coefficient, 8 -- ( -- times the exponent of The next coefficient -- ( -- is 28 28 The next -- ( -- is 56 56 We have now come to the point of symmetry. For, the coefficient of Here is the complete expansion:
Example 6. Write the 5th term in the expansion of ( The index k -- the exponent of Thus in the
Pascal's triangle This triangular array is called Pascal's Triangle. Each row gives the combinatorial numbers, which are the binomial coefficients. That is, the row 1 2 1 are the combinatorial numbers To construct the triangle, write 1, and below it write 1 1. Begin and end each successive row with 1. To construct the intervening numbers, add the two numbers immediately above. Thus to construct the third row, begin it with 1, then add the two numbers immediately above: 1 + 1. Write 2. Finish the row with 1. To construct the next row, begin it with 1, and add the two numbers immediately above: 1 + 2. Write 3. Again, add the two numbers immediately above: 2 + 1 = 3. Finish the row with 1.
Example 8. Expand ( ^{6}. 1 6 15 20 15 6 1. In the binomial,
Example 9. Expand ( ^{3}.
Problem 1. In the expansion of ( where is the symbol for the binomial coefficient. The binomial theorem is the statement that = ? To see the answer, pass your mouse over the colored area.
The combinatorial number,
Problem 2. Use factorials to write the general term in the expansion of (
Problem 3. a) Calculate the coefficient of
b) The coefficient of which other term is the same?
c) In the expansion of ( Problem 4. Calculate the coefficient of a) b) c) d) e) f)
Problem 5. Write the first four terms of (
Problem 6. Compute the first four terms of each of the following. a) ( b) (
Problem 7. Consider the expansion of ( a) What is the exponent of b) What is the exponent of c) In the d) In the e) Write the fourth term, with its coefficient.
4,060 Problem 8. Calculate each of the following. a) The third term of ( b) The fifth term of ( c) The tenth term of (
Problem 9. Use Pascal's triangle to expand the following. a) ( b) ( c) ( d) ( e) ( f) ( g) (2 h) (1 −
= 1 − 7 In the following Topic we will explain why the binomial coefficients are the combinatorial numbers. That will constitute a proof of the binomial theorem. Next Topic: Multiplication of sums Please make a donation to keep TheMathPage online. Copyright © 2016 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |