Question

How do I find the constant term of a binomial expansion?

Answer 1

Hint: In the above question, you were asked to find the constant term of binomial expression. According to the formula of the binomial theorem that is ${{(x+y)}^{n}}$ , the term ${{y}^{n}}$ is always constant. You will see how this is a constant term. So, let us see how we can solve this problem.

Complete step by step solution:
We will see what do we get from the expansion of the binomial formula which is ${{(x+y)}^{n}}$
 $\Rightarrow {{(x+y)}^{n}}=(\begin{matrix}
   n \\
   0 \\
\end{matrix}).{{x}^{n}}+(\begin{matrix}
   n \\
   1 \\
\end{matrix}).{{x}^{n-1}}.{{y}^{1}}....+(\begin{matrix}
   n \\
   k \\
\end{matrix}).{{x}^{n-k}}.{{y}^{k}}+....+(\begin{matrix}
   n \\
   n \\
\end{matrix}).{{y}^{n}}=\sum\limits_{k=0}^{n}{.(\begin{matrix}
   n \\
   k \\
\end{matrix}).{{x}^{n-k}}.{{y}^{k}}}$
where $x,y\in \mathbb{R},$ $k,n\in \mathbb{N}$ and $(\begin{matrix}
   n \\
   k \\
\end{matrix})$ denotes combinations of n things taken k at a time. So we have 2 cases
1st case: When the terms of the binomial are a constant and a variable like
 $\Rightarrow {{(x+c)}^{n}}=(\begin{matrix}
   n \\
   0 \\
\end{matrix})\cdot {{x}^{n}}+(\begin{matrix}
   n \\
   1 \\
\end{matrix})\cdot {{x}^{n-1}}\cdot {{c}^{1}}+...+(\begin{matrix}
   n \\
   k \\
\end{matrix})\cdot {{x}^{n-k}}\cdot {{c}^{k}}+...+(\begin{matrix}
   n \\
   n \\
\end{matrix})\cdot {{c}^{n}}$
Here the constant term is $(\begin{matrix}
   n \\
   n \\
\end{matrix})\cdot {{c}^{n}}$ and its product is also constant.
2nd Case: When the terms of the binomial are a variable and the ratio of that variable like
 $\Rightarrow (\begin{matrix}
   n \\
   k \\
\end{matrix})\cdot {{x}^{n-k}}.{{(\dfrac{c}{x})}^{k}}=(\begin{matrix}
   n \\
   k \\
\end{matrix})\cdot {{x}^{n-k}}\cdot {{c}^{k}}.\dfrac{1}{{{x}^{k}}}=((\begin{matrix}
   n \\
   n \\
\end{matrix})\cdot {{c}^{k}}).\dfrac{{{x}^{n-k}}}{{{x}^{k}}}=((\begin{matrix}
   n \\
   k \\
\end{matrix})\cdot {{c}^{k}}).{{x}^{n-2k}}$
Therefore, the middle term is constant in this case that is $k=\dfrac{n}{2}$.

So, the constant term in a binomial expression which is ${{(x+y)}^{n}}$ is ${{y}^{n}}$.

Note:
For the above solution, there was one more case but it has no constant term. The binomial expression for the third case is ${{(x+y)}^{n}}$. We will study the details of the binomial theorem in the coming lessons.