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 ${(x+y)}^n$
 $\Rightarrow {(x+y)}^n=(\begin{array}{c}n\\ 0\end{array}).x^n+(\begin{array}{c}n\\ 1\end{array}).x^{n-1}.y^1....+(\begin{array}{c}n\\ k\end{array}).x^{n-k}.y^k+....+(\begin{array}{c}n\\ n\end{array}).y^n=\sum _{k=0}^n.(\begin{array}{c}n\\ k\end{array}).x^{n-k}.y^k$
where $x,y\in \mathbb{R},$ $k,n\in \mathbb{N}$ and $(\begin{array}{c}n\\ k\end{array})$ 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
 $\Rightarrow {(x+c)}^n=(\begin{array}{c}n\\ 0\end{array})\cdot x^n+(\begin{array}{c}n\\ 1\end{array})\cdot x^{n-1}\cdot c^1+...+(\begin{array}{c}n\\ k\end{array})\cdot x^{n-k}\cdot c^k+...+(\begin{array}{c}n\\ n\end{array})\cdot c^n$
Here the constant term is $(\begin{array}{c}n\\ n\end{array})\cdot c^n$ and its product is also constant.
2nd Case: When the terms of the binomial are a variable and the ratio of that variable like
 $\Rightarrow (\begin{array}{c}n\\ k\end{array})\cdot x^{n-k}.{({\displaystyle \frac{c}{x}})}^k=(\begin{array}{c}n\\ k\end{array})\cdot x^{n-k}\cdot c^k.{\displaystyle \frac{1}{x^k}}=((\begin{array}{c}n\\ n\end{array})\cdot c^k).{\displaystyle \frac{x^{n-k}}{x^k}}=((\begin{array}{c}n\\ k\end{array})\cdot c^k).x^{n-2k}$
Therefore, the middle term is constant in this case that is $k={\displaystyle \frac{n}{2}}$.

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.