The expansion of a binomial is given by the Binomial Theorem:
#(x+y)^n=( (n), (0) )*x^n+( (n), (1) )*x^(n-1)*y^1+...+( (n), (k) )*x^(n-k)*y^k+...+( (n), (n) )*y^n = sum_(k=0)^n*( (n), (k) )*x^(n-k)*y^k #
where #x, y in RR#, #k, n in NN#, and #( (n), (k) )# denotes combinations of #n# things taken #k# at a time.
# ( (n), (k) )*x^(n-k)*y^k # is the general term of the binomial expansion.
We also have the formula: #( (n), (k) )=(n!)/(k!*(n-k)!)#, where #k! = 1*2*...*k#
We have three cases:
Case 1: If the terms of the binomial are a variable and a constant #(y=c#, where #c# is a constant), we have #(x+c)^n=( (n), (0) )*x^n+( (n), (1) )*x^(n-1)*c^1+...+( (n), (k) )*x^(n-k)*c^k+...+( (n), (n) )*c^n #
We can see that the constant term is the last one: #( (n), (n) )*c^n#
(as #( (n), (n) )# and #c^n# are constant, their product is also a constant).
Case 2: If the terms of the binomial are a variable and a ratio of that variable (#y=c/x#, where #c# is a constant), we have:
# (x+c/x)^n=( (n), (0) )*x^n + ( (n), (1) )*x^(n-1)*(c/x)^1+...+( (n), (k) )*x^(n-k)*(c/x)^k+...+( (n), (n) )*(c/x)^n #
This time, we see that the constant term is not to be found at the extremities of the binomial expansion. So, we should have a look at the general term and try to find out when it becomes a constant:
# ( (n), (k) )*x^(n-k)*(c/x)^k=( (n), (k) )*x^(n-k)*c^k*1/x^k = (( (n), (n) )*c^k)*(x^(n-k))/x^k = (( (n), (k) )*c^k)*x^(n-2k) #.
We can see that the general term becomes constant when the exponent of variable #x# is #0#. Therefore, the condition for the constant term is: #n-2k=0 rArr# #k=n/2# . In other words, in this case, the constant term is the middle one (#k=n/2#).
Case 3: If the terms of the binomial are two distinct variables #x# and #y#, such that #y# cannot be expressed as a ratio of #x#, then there is no constant term . This is the general case #(x+y)^n#
