Question

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

Answer 1

Hint: Here we can write the general form of $\left(r+1\right)\text{th term}$ and this is for the expansion ${\left(a+b\right)}^n$ which is:
$T_{r+1}={}^n{C}_r{\left(a\right)}^{n-r}{\left(b\right)}^r$.
So we can simplify this and put the degree of the variable term as zero to get the value of $r$ and therefore we will get the constant term of that binomial expansion.

Complete step by step solution:
Here we are given to find the constant term of any expansion using the binomial theorem.
We can write the general form of $\left(r+1\right)\text{th term}$ and this is for the expansion ${\left(a+b\right)}^n$which is:
$T_{r+1}={}^n{C}_r{\left(a\right)}^{n-r}{\left(b\right)}^r$.
So we can simplify this and put the degree of the variable term as zero to get the value of $r$ and therefore we will get the constant term of that binomial expansion.
This can be made clear with one example:
For example: We need to find the constant term of the expansion ${\left(2x+3\right)}^3$
So we know that we need to use binomial expansion over here.
We know that general expansion of the term ${\left(a+b\right)}^n$ is $T_{r+1}={}^n{C}_r{\left(a\right)}^{n-r}{\left(b\right)}^r$ which tells us that it is $\left(r+1\right)\text{th term}$.
So we can compare ${\left(a+b\right)}^n$ with ${\left(2x+3\right)}^3$ and we will get that:
$$\begin{gathered} a=2x \\ b=3 \\ n=3 \end{gathered}$$
Now we can write the general form of $\left(r+1\right)\text{th term}$ of the above term as:
$T_{r+1}={}^3{C}_r{\left(2x\right)}^{3-r}{\left(3\right)}^r$
We can simplify it and write it as:
$T_{r+1}={}^3{C}_r{\left(2\right)}^{3-r}{\left(3\right)}^r{\left(x\right)}^{3-r}$
So for the constant term we must not have any term containing $x$ so we can write the degree of $x$ as zero and we will get:
$$\begin{gathered} 3-r=0 \\ r=3 \end{gathered}$$
Hence now we can say that $\left(r+1\right)=4\text{th term}$ is the constant term and it is:
$T_4={}^3{C}_3{\left(2\right)}^{3-3}{\left(3\right)}^3=\left(1\right)\left(1\right)\left(3\right)\left(3\right)\left(3\right)=27$

Hence the constant term is $27$.

Note: Here the student can also be given the constant term and told to find the value of any unknown variable also. So a similar process needs to be followed and then we need to just apply the general formula and compare.