Question

What is the binomial expansion of \[{{\left( 2x+3 \right)}^{5}}\]?

Answer 1

Hint: For solving this question you should know about the binomial of two terms or more terms. A binomial is a polynomial with two terms. And to write the expansion of any two terms is known as the binomial expansion. And this expansion is always in an order that order or manner is known as Binomial Theorem.

Complete step-by-step answer:
According to our question we have to find the binomial expansion of \[{{\left( 2x+3 \right)}^{5}}\].
So, as if we know that a polynomial of two terms is known as a binomial. And if we want to expand these terms then these will expand in a fixed manner we can see it from this example-
Eg. (1) What happens if we multiply a binomial a + b by itself.
 So, here a + b is a binomial (the two terms are a and b)
Let us multiply \[\left( a+b \right)\] by itself:
\[\Rightarrow \left( a+b \right)\left( a+b \right)={{a}^{2}}+2ab+{{b}^{2}}\]
Now, again we multiply it with \[\left( a+b \right)\]:
\[\Rightarrow \left( {{a}^{2}}+2ab+{{b}^{2}} \right)\left( a+b \right)={{a}^{3}}+3{{a}^{2}}b+3a{{b}^{2}}+{{b}^{3}}\]
This calculation will get longer and longer as we go.
And here some kind of pattern developing which is \[{{\left( a+b \right)}^{n}}=\sum\limits_{k=0}^{n}{\left( \begin{matrix}
   n \\
   k \\
\end{matrix} \right){{a}^{n-k}}{{b}^{k}}}\].
And this pattern is known as the Binomial theorem.
So, according to the binomial Theorem if we expand to \[{{\left( a+b \right)}^{5}}\] the:
\[{{\left( a+b \right)}^{5}}=\sum\limits_{k=0}^{5}{\left( \begin{matrix}
   n \\
   k \\
\end{matrix} \right){{a}^{5-k}}{{b}^{k}}}\]
\[\begin{align}
  & {{\left( a+b \right)}^{5}}=\left( \begin{matrix}
   5 \\
   0 \\
\end{matrix} \right){{a}^{5}}{{b}^{0}}+\left( \begin{matrix}
   5 \\
   1 \\
\end{matrix} \right){{a}^{4}}b+\left( \begin{matrix}
   5 \\
   2 \\
\end{matrix} \right){{a}^{3}}{{b}^{2}}+\left( \begin{matrix}
   5 \\
   3 \\
\end{matrix} \right){{a}^{2}}{{b}^{3}}+\left( \begin{matrix}
   5 \\
   4 \\
\end{matrix} \right)a{{b}^{4}}+\left( \begin{matrix}
   5 \\
   5 \\
\end{matrix} \right){{a}^{0}}{{b}^{5}} \\
 & \left( \begin{matrix}
   n \\
   r \\
\end{matrix} \right)={}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!} \\
\end{align}\]
Let \[a=2x\] and \[b=3\]
So, \[{{\left( a+b \right)}^{5}}=1.{{a}^{5}}+5{{a}^{4}}b+10{{a}^{3}}{{b}^{2}}+10{{a}^{2}}{{b}^{3}}+5a{{b}^{4}}+{{b}^{5}}\]
Now substituting the values in this:
\[\begin{align}
  & \Rightarrow {{\left( 2x+3 \right)}^{5}}={{\left( 2x \right)}^{5}}+5{{\left( 2x \right)}^{4}}\left( 30 \right)+10{{\left( 2x \right)}^{3}}{{\left( 3 \right)}^{2}}+10{{\left( 2x \right)}^{2}}{{\left( 3 \right)}^{3}}+5\left( 2x \right){{\left( 3 \right)}^{4}}+{{\left( 3 \right)}^{5}} \\
 & \Rightarrow {{\left( 2x+3 \right)}^{5}}=32{{x}^{5}}+240{{x}^{4}}+720{{x}^{3}}+1080{{x}^{2}}+810x+243 \\
\end{align}\]
So, the binomial expansion of \[{{\left( 2x+3 \right)}^{5}}\] is \[32{{x}^{5}}+240{{x}^{4}}+720{{x}^{3}}+1080{{x}^{2}}+810x+243\].

Note: During solving the questions of Binomial expansions we have to be careful for the powers of the terms because the powers according to the binomial theorem will be an important step is the main point of this question. And then substitute values carefully in this. And solve all calculations very carefully.