Question

Find the term independent of ‘x’ in the expansion of $ {\left( {4{x^3} + \dfrac{7}{{{x^2}}}} \right)^{14}} $ .

Answer 1

Hint: The given question requires us to find the term independent of x in the binomial expansion $ {\left( {4{x^3} + \dfrac{7}{{{x^2}}}} \right)^{14}} $ . Binomial theorem helps us to expand the powers of binomial expressions easily and can be used to solve the given problem. We must know the formulae of combinations and factorials for solving the given question using the binomial theorem. To solve the problem, we find the general term of the binomial expansion and then equate the power of x as zero.

Complete step by step solution:
So, we are given the binomial expression $ {\left( {4{x^3} + \dfrac{7}{{{x^2}}}} \right)^{14}} $ .
So, using the binomial theorem, the binomial expansion of \[{\left( {x + y} \right)^n}\] is $ \sum\nolimits_{r = 0}^n {\left( {^n{C_r}} \right){{\left( x \right)}^{n - r}}{{\left( y \right)}^r}} $
So, the binomial expansion of $ {\left( {4{x^3} + \dfrac{7}{{{x^2}}}} \right)^{14}} $ is \[\sum\nolimits_{r = 0}^{14} {\left( {^{14}{C_r}} \right){{\left( {4{x^3}} \right)}^{14 - r}}{{\left( {\dfrac{7}{{{x^2}}}} \right)}^r}} \] .
Now, we know that the general term of the binomial expansion of the expression $ {\left( {4{x^3} + \dfrac{7}{{{x^2}}}} \right)^{14}} $ is \[^{14}{C_r}{\left( {4{x^3}} \right)^{14 - r}}{\left( {\dfrac{7}{{{x^2}}}} \right)^r}\].
Now, we simplify this expression and compute the power of x,
\[{ \Rightarrow ^{14}}{C_r}{\left( 4 \right)^{14 - r}}{\left( 7 \right)^r}{x^{3\left( {14 - r} \right) - 2r}}\]
Simplifying the expression further, we get,
\[{ \Rightarrow ^{14}}{C_r}{\left( 4 \right)^{14 - r}}{\left( 7 \right)^r}{x^{42 - 5r}}\]
Now, we equate the power of x to zero so as to find the term independent of x.
\[42 - 5r = 0\]
Now, shifting the terms in the equation to find the value of r, we get,
\[ \Rightarrow 5r = 42\]
\[ \Rightarrow r = \dfrac{{42}}{5}\]
Since the value of r comes out to be a fraction and not a positive integer. So, we cannot substitute the obtained value of r in the general term of the binomial expansion as we cannot put fractional values in \[^n{C_r}\] formula.
Hence, the term independent of x in the expansion of $ {\left( {4{x^3} + \dfrac{7}{{{x^2}}}} \right)^{14}} $ does not exists.

Note: The given problem can be solved by various methods. The easiest way is to apply the concepts of Binomial theorem as it is very effective in finding the binomial expansion. But the problem can also be solved by actually calculating the power of the binomial expansion given, though it is a tedious task to calculate such a high power of a binomial polynomial. This can verify the final answer of the problem.