Question

How do you use the binomial series to expand \[{{\left( 1+2x \right)}^{5}}\]?

Answer 1

Hint: The term of the form \[{{\left( a+b \right)}^{n}}\] is called a binomial term. The general form of the expansion of the binomial term is \[\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}{{a}^{r}}{{b}^{n-r}}}\]. We can find the binomial series by substituting the values of a, b, and n in this summation. It should be noted that here \[n\] is a positive integer.

Complete step-by-step answer:
We are asked to expand the binomial term \[{{\left( 1+2x \right)}^{5}}\]. Comparing with the general binomial term \[{{\left( a+b \right)}^{n}}\], we get \[a=1,b=2x\And n=5\]. We know the general form of the binomial series is \[\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}{{a}^{r}}{{b}^{n-r}}}\]. We can expand the given binomial term by substituting the value of variables in the general form, as follows
\[\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}{{a}^{r}}{{b}^{n-r}}}\]
\[\Rightarrow \sum\limits_{r=0}^{5}{^{5}{{C}_{r}}{{\left( 1 \right)}^{r}}{{\left( 2x \right)}^{5-r}}}\]
Expanding the above summation, we get
\[{{\Rightarrow }^{5}}{{C}_{0}}{{\left( 1 \right)}^{0}}{{\left( 2x \right)}^{5-0}}{{+}^{5}}{{C}_{1}}{{\left( 1 \right)}^{1}}{{\left( 2x \right)}^{5-1}}{{+}^{5}}{{C}_{2}}{{\left( 1 \right)}^{2}}{{\left( 2x \right)}^{5-2}}{{+}^{5}}{{C}_{3}}{{\left( 1 \right)}^{3}}{{\left( 2x \right)}^{5-3}}{{+}^{5}}{{C}_{4}}{{\left( 1 \right)}^{4}}{{\left( 2x \right)}^{5-4}}{{+}^{5}}{{C}_{5}}{{\left( 1 \right)}^{5}}{{\left( 2x \right)}^{5-5}}\]
 We know that \[^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\], using this to simplify the above series we get
\[^{5}{{C}_{0}}=\dfrac{5!}{0!\left( 5-0 \right)!}=1\]
\[^{5}{{C}_{1}}=\dfrac{5!}{1!\left( 5-1 \right)!}=5\]
\[^{5}{{C}_{2}}=\dfrac{5!}{2!\left( 5-2 \right)!}=10\]
\[^{5}{{C}_{3}}=\dfrac{5!}{3!\left( 5-3 \right)!}=10\]
\[^{5}{{C}_{4}}=\dfrac{5!}{4!\left( 5-4 \right)!}=5\]
\[^{5}{{C}_{5}}=\dfrac{5!}{5!\left( 5-5 \right)!}=1\]
Substituting the values of the above coefficients, we get
\[\Rightarrow 1{{\left( 2x \right)}^{5}}+5{{\left( 2x \right)}^{4}}+10{{\left( 2x \right)}^{3}}+10{{\left( 2x \right)}^{2}}+5{{\left( 2x \right)}^{1}}+1{{\left( 2x \right)}^{0}}\]
Simplifying the above series, we get
\[\Rightarrow 32{{x}^{5}}+80{{x}^{4}}+80{{x}^{3}}+40{{x}^{2}}+10x+1\]
Is the expansion of the given binomial term.

Note: We can use more special binomial expansions to expand the series. Here one of the terms inside the bracket is 1. Hence, 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 this series, we have to substitute \[2x\] at the place of \[x\]. 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.