Question

What is the Binomial expansion of ${{\left( 2x+3 \right)}^{4}}$?

Answer 1

To obtain the binomial expansion of the expression given use Binomial Theorem. Firstly we will use the binomial theorem to expand the power of the expression. Then by using factorial and combination formulas we will simplify the equation further. Finally we will solve the equation and get the desired answer.

Complete step by step solution:
We have to find Binomial Expansion of the binomial expression:
${{\left( 2x+3 \right)}^{4}}$……$\left( 1 \right)$
The Binomial Theorem state that if we have to solve the below expression the formula used is:
${{\left( x+y \right)}^{n}}=\sum\nolimits_{r=0}^{n}{{{x}^{n-r}}{{y}^{r}}{}^{n}{{C}_{r}}}$……$\left( 2 \right)$
Where,
$x,y=$ Any real numbers/variables
$n=$ Any positive integer
$r=$ Natural number that takes value from 0 to $n$
On comparing equation (1) and (2) we get the values as:
$n=4$
$x=2x$
$y=3$
So equation (1) can be written as below:
${{\left( 2x+3 \right)}^{4}}=\sum\nolimits_{r=0}^{4}{{{\left( 2x \right)}^{4-r}}{{\left( 3 \right)}^{r}}{}^{4}{{C}_{r}}}$
On expanding the above equation and simplifying it we get,
$\begin{aligned} & \Rightarrow \left( {{\left( 2x \right)}^{4-0}}{{\left( 3 \right)}^{0}}{}^{4}{{C}_{0}} \right)+\left( {{\left( 2x \right)}^{4-1}}{{\left( 3 \right)}^{1}}{}^{4}{{C}_{1}} \right)+\left( {{\left( 2x \right)}^{4-2}}{{\left( 3 \right)}^{2}}{}^{4}{{C}_{2}} \right)+\left( {{\left( 2x \right)}^{4-3}}{{\left( 3 \right)}^{3}}{}^{4}{{C}_{3}} \right)+\left( {{\left( 2x \right)}^{4-4}}{{\left( 3 \right)}^{4}}{}^{4}{{C}_{4}} \right) \\\ & \Rightarrow \left( {{\left( 2x \right)}^{4}}\times 1\times {}^{4}{{C}_{0}} \right)+\left( {{\left( 2x \right)}^{3}}\times 3\times {}^{4}{{C}_{1}} \right)+\left( {{\left( 2x \right)}^{2}}\times {{\left( 3 \right)}^{2}}\times {}^{4}{{C}_{2}} \right)+\left( {{\left( 2x \right)}^{1}}\times {{\left( 3 \right)}^{3}}\times {}^{4}{{C}_{3}} \right)+\left( 1\times {{\left( 3 \right)}^{4}}\times {}^{4}{{C}_{4}} \right) \\\ \end{aligned}$
Now we will use combination formula in above value which is given as below:
${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$
Where $n!=n\times \left( n-1 \right)\times \left( n-2 \right)\times .........\times 1$
Using above formula in our equation and simplifying we get,
$\begin{aligned} & \Rightarrow \left( {{\left( 2x \right)}^{4}}\times 1\times \dfrac{4!}{0!\left( 4-0 \right)!} \right)+\left( {{\left( 2x \right)}^{3}}\times 3\times \dfrac{4!}{1!\left( 3-1 \right)!} \right)+\left( {{\left( 2x \right)}^{2}}\times {{\left( 3 \right)}^{2}}\times \dfrac{4!}{2!\left( 4-2 \right)!} \right)+\left( {{\left( 2x \right)}^{1}}\times {{\left( 3 \right)}^{3}}\times \dfrac{4!}{3!\left( 4-3 \right)!} \right)+\left( 1\times {{\left( 3 \right)}^{4}}\times \dfrac{4!}{4!\left( 4-4 \right)!} \right) \\\ & \Rightarrow \left( {{\left( 2x \right)}^{4}}\times 1\times \dfrac{4!}{1\times 4!} \right)+\left( {{\left( 2x \right)}^{3}}\times 3\times \dfrac{4!}{1!\times 3!} \right)+\left( {{\left( 2x \right)}^{2}}\times {{\left( 3 \right)}^{2}}\times \dfrac{4!}{2!\times 2!} \right)+\left( {{\left( 2x \right)}^{1}}\times {{\left( 3 \right)}^{3}}\times \dfrac{4!}{3!\times 1!} \right)+\left( 1\times {{\left( 3 \right)}^{4}}\times \dfrac{4!}{4!\times 0!} \right) \\\ & \Rightarrow \left( {{\left( 2x \right)}^{4}}\times 1\times 1 \right)+\left( {{\left( 2x \right)}^{3}}\times 3\times 4 \right)+\left( {{\left( 2x \right)}^{2}}\times {{\left( 3 \right)}^{2}}\times 6 \right)+\left( {{\left( 2x \right)}^{1}}\times {{\left( 3 \right)}^{3}}\times 4 \right)+\left( 1\times {{\left( 3 \right)}^{4}}\times 1 \right) \\\ & \Rightarrow 16{{x}^{4}}+96{{x}^{3}}+216{{x}^{2}}+216x+81 \\\ \end{aligned}$

Hence binomial expansion of ${{\left( 2x+3 \right)}^{4}}$ is $16{{x}^{4}}+96{{x}^{3}}+216{{x}^{2}}+216x+81$.

Note: To solve problems of binomial expansion Binomial theorem is widely used although the calculation part does get complicated but it is still preferred. A binomial expression is an algebraic expression which has two terms connected by plus or minus sign. The coefficient of each term is known as binomial coefficient. The two formulas that one needs to know for using Binomial theorem is Combination and factorial. The coefficient of the binomial terms can be found by another method known as Pascal’s method.