Question

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

Answer 1

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.