Question

How do I find missing values in binomial expansions?

Answer 1

Hint: The term of the form \[{{\left( a+b \right)}^{n}}\] is called a binomial term. In the expansion of this binomial, there are total n+1 terms. The (r+1)th term of the expansion of the binomial expansion is \[^{n}{{C}_{r}}{{a}^{r}}{{b}^{n-r}}\]. We can find the binomial term by substituting the values of a, b, and n in this general term. It should be noted that here \[n\] is a positive integer. To solve the given problem, we will assume a binomial expansion and find its missing term using the information given above.

Complete step by step solution:
Let’s assume we are given the expansion of the binomial as \[{{\left( 1+2x \right)}^{a}}=b+8x+c{{x}^{2}}+32{{x}^{3}}+16{{x}^{4}}\]. We need to find the missing terms.
As the highest power in the expansion on the right-hand side is 4. The power of the term on the left-hand side must be also 4. Hence, we get \[a=4\].
The first term on the right side is a constant term. The only constant term we can get from the binomial on the left side is 1. Hence, we get \[b=1\].
We want to find the coefficient of \[{{x}^{2}}\]. It is third term on the right side, so its coefficient will be \[^{4}{{C}_{2}}\times {{2}^{2}}=24\]. Hence, we get \[c=24\].
In this way, using the binomial and its expansion we can find their missing terms.

Note: We can use more special binomial expansions to expand the series. If one of the terms inside the bracket is 1. then, we can use the expansion of \[{{\left( 1+x \right)}^{n}}\] whose general form of expansion is \[\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}{{x}^{r}}}\]. For these types of problems, these expansions are very important and should be remembered.
We can use these expansions only when \[n\] is a positive integer. For cases when the \[n\] is a non-positive integer, we need to use different types of expansions.