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:
$
  a = 2x \\
  b = 3 \\
  n = 3 \\
 $
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:
$
  3 - r = 0 \\
  r = 3 \\
 $
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.