Question

If \[x=2+\dfrac{{{2}^{1}}}{3}+\dfrac{{{2}^{2}}}{3}\], find the value of \[-\left( {{x}^{3}}-6{{x}^{2}}+6x \right)\]
(a) 2
(b) 4
(c) 8
(d) 6

Answer 1

Hint: In this question, we first need to subtract 2 from both sides of the given x value. Then do cubing on both sides. Now, expand using the factorisation of polynomials formula given by \[{{\left( x-a \right)}^{3}}={{x}^{3}}-3{{x}^{2}}a+3x{{a}^{2}}-{{a}^{3}}\]and then simplify accordingly to get the result.

Complete step-by-step answer:
Now, from the given conditions in the question we have
\[x=2+\dfrac{{{2}^{1}}}{3}+\dfrac{{{2}^{2}}}{3}\]
Now, let us subtract 2 on both sides of the above expression
\[\Rightarrow x-2=\dfrac{{{2}^{1}}}{3}+\dfrac{{{2}^{2}}}{3}\]
Let us now do the cubing on both sides of the above expression
\[\Rightarrow {{\left( x-2 \right)}^{3}}={{\left( \dfrac{{{2}^{1}}}{3}+\dfrac{{{2}^{2}}}{3} \right)}^{3}}\]
Let us first simplify the right hand side of the above expression
\[\Rightarrow {{\left( x-2 \right)}^{3}}={{\left( \dfrac{2+4}{3} \right)}^{3}}\]
Now, on further simplification of the above expression we get,
\[\Rightarrow {{\left( x-2 \right)}^{3}}={{\left( \dfrac{6}{3} \right)}^{3}}\]
Now, this can also be written in the simplified form as follows
\[\Rightarrow {{\left( x-2 \right)}^{3}}={{\left( 2 \right)}^{3}}\]
As we already know from the factorisation of the polynomials that
\[{{\left( x-a \right)}^{3}}={{x}^{3}}-3{{x}^{2}}a+3x{{a}^{2}}-{{a}^{3}}\]
Now, let us expand the left hand side of the expression using the above formula
Now, on comparison with the formula we have
\[a=2\]
Now, on substituting the value of a in the respective formula we get,
\[{{\left( x-2 \right)}^{3}}={{x}^{3}}-3\times {{x}^{2}}\times 2+3\times x\times {{2}^{2}}-{{2}^{3}}\]
Now, substituting this expansion in the above expression we get,
\[\Rightarrow {{x}^{3}}-3\times {{x}^{2}}\times 2+3\times x\times {{2}^{2}}-{{2}^{3}}={{\left( 2 \right)}^{3}}\]
Now, this can also be written as
\[\Rightarrow {{x}^{3}}-6{{x}^{2}}+12x-{{2}^{3}}={{2}^{3}}\]
Now, on rearranging the terms we get,
\[\Rightarrow {{x}^{3}}-6{{x}^{2}}+12x=16\]
Now, we have to find the value of the following expression
\[\Rightarrow {{x}^{3}}-6{{x}^{2}}+6x\]
Let us now add and subtract the term \[6x\]form the above expression
\[\Rightarrow {{x}^{3}}-6{{x}^{2}}+12x-6x\]
Let us now substitute the value obtained in the above calculation
\[\Rightarrow 16-6x\]
Let us now substitute the value of x in the above expression
\[\Rightarrow 16-6\left( 2+\dfrac{{{2}^{1}}}{3}+\dfrac{{{2}^{2}}}{3} \right)\]
Now, on multiplying the respective terms we get,
\[\Rightarrow 16-6\times 2-\dfrac{6\times 2}{3}-\dfrac{6\times 4}{3}\]
Now, the above expression can also be written as
\[\Rightarrow 16-12-4-8\]
Now, on further simplification we get,
\[\Rightarrow {{x}^{3}}-6{{x}^{2}}+6x=-8\]
Now, on interchanging the negative sign to the other side we get,
\[\therefore -\left( {{x}^{3}}-6{{x}^{2}}+6x \right)=8\]
Hence, the correct option is (c).

Note:Instead of subtracting 2 on both sides and then cubing we can also solve it by simplifying the value of the given x and then substitute that value in the given expression. Both the methods given the same result.It is important to note that we can also convert the given expression in terms of x and then substitute its value to simplify further. It is also to be noted that while expanding we should not neglect any of the term.