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

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Oct 1, 2015

Let ${\left(2 x + 3\right)}^{3}$ 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 ${\left(2 x\right)}^{3 - r}$ ${3}^{r}$

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.

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