How do I use the binomial theorem to find the constant term?

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Oct 1, 2015

Let #(2x+3) ^3# be a given binomial.

From the binomial expression, write down the general term. Let this term be the r+1 th term. Now simplify this general term. If this general term is a constant term, then it should not contain the variable x.
Let us write the general term of the above binomial.
#T_(r+1)# = #"" ^3 C_r# #(2x)^(3-r)# #3^r#

simplifying, we get, #T_(r+1)#= #"" ^3 C_r# #2^(3-r)# #3^r# #x^(3-r)#

Now for this term to be the constant term, #x^(3-r)# should be equal to 1.
Therefore, #x^(3-r)#= #x^0#
=> 3-r =0
=> r=3

Thus, the fourth term in the expansion is the constant term. By putting r=3 in the general term, we will get the value of the constant term.

cliffsnotes.com

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