How can you use binomial expansion to expand the expression \[{(2x + 1)^4}\] ?

Hint: In this problem, we need to expand the given \[{(2x + 1)^4}\] by means of binomial theorem. The expression contains two unlike terms \[2x\,\,and\,1\] .Hence it’s a binomial term. To evaluate the higher powers of a binomial expansion normal expansion would be a tedious task. So we use the conventional binomial expansion.

Complete step-by-step answer:
We need to first express the general formula of binomial expression. Then we need to superimpose the given expression into that form. We need to identify the different variables and put the values to obtain the desired polynomial form.
Any binomial term of the form \[{(a + b)^n}\] can be expressed as \[{(a + b)^n}\, = \,\sum\limits_{k = 0}^n {{}^n{C_k} \cdot ({a^{n - k}}{b^k}).} \]
It can also be written as: \[{(a + b)^n}\, = {}^n{C_0}{a^n}{b^0} + {}^n{C_1}{a^{n - 1}}{b^1} + {}^n{C_2}{a^{n - 2}}{b^2} + \cdot \cdot \cdot + {}^n{C_n}{a^0}{b^n}\] .
Now, expanding the expression, \[{}^n{C_{r\,}}\, = \,\dfrac{{n!}}{{(n - r)!r!}}\]
We can rewrite the binomial expansion as:
 \[{(a + b)^n} = {a^n} + n{a^{n - 1}}b + \dfrac{{n(n - 1)}}{{2!}}{a^{n - 2}}{b^2} + \dfrac{{n(n - 1)(n - 2)}}{{3!}}{a^{n - 3}}{b^3} + \cdot \cdot \cdot + {b^n}.\]
Now, according to the question \[a = 2x\,\,;\,\,b = 1\] . Putting the values of a a and b in the obtained binomial expansion we get;
 \[
 {(2x + 1)^4} = \dfrac{{4!}}{{4!0!}}{(2x)^4}{(1)^0} + \dfrac{{4!}}{{3!1!}}{(2x)^3}{(1)^1} + \dfrac{{4!}}{{2!2!}}{(2x)^2}{(1)^2} + \dfrac{{4!}}{{1!3!}}{(2x)^1}{(1)^3} + \dfrac{{4!}}{{0!4!}}{(2x)^0}{(1)^4} \\
   \Rightarrow \,\,\,{(2x + 1)^4} = 1{(2x)^4} + 4{(2x)^3} + 6{(2x)^2} + 4{(2x)^1} + 1{(2x)^0} \\
   \Rightarrow \,\,\,{(2x + 1)^4} = 16{x^4} + 32{x^3} + 24{x^2} + 8x + 1 \;
 \]
So the expanded polynomial we get after the binomial expansion of the expression \[{(2x + 1)^4}\] is \[\,\,{(2x + 1)^4} = 16{x^4} + 32{x^3} + 24{x^2} + 8x + 1\]
So, the correct answer is “ \[\,\,{(2x + 1)^4} = 16{x^4} + 32{x^3} + 24{x^2} + 8x + 1\] ”.

Note: Based on the properties of the binomial theorem, the binomial formula is valid for all positive integer values of n. The total number of terms in the binomial expansion of \[{(a + b)^n}\] is \[(n + 1)\]. It is also to be noted that the sum of exponents of \[a\] and \[b\] in each term of the expansion is \[n\]. While expanding the binomial coefficient we can observe that the exponents of \[a\] increases by 1 while the exponents of \[b\] decrease by 1.