Use the General Term \displaystyle {{T}_{{r+1}}}=\left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right){{a}^{{n-r}}}{{b}^{r}}  where n=10,  a=\displaystyle {\frac{3}{x}}  and  b=\displaystyle {-{{x}^{2}}} to find if there is a term that is independent of x (i.e. x0). If there is, the “r” value will be either a 0 or a positive whole number.

\displaystyle \left( {\begin{array}{*{20}{c}} {10} \\ r \end{array}} \right){{\left( {\frac{3}{x}} \right)}^{{10-r}}}{{(-{{x}^{2}})}^{r}}

\displaystyle \left( {\begin{array}{*{20}{c}} {10} \\ r \end{array}} \right){{\left( 3 \right)}^{{10-r}}}{{\left( {\frac{1}{x}} \right)}^{{10-r}}}{{(-1)}^{r}}{{({{x}^{2}})}^{r}}

Compare the x terms and equate it to x to the power of zero which is the term independent of x.

\displaystyle {{\left( {\frac{1}{x}} \right)}^{{10-r}}}{{({{x}^{2}})}^{r}}={{x}^{0}}

\displaystyle {{\left( {{{x}^{{-1}}}} \right)}^{{10-r}}}\times {{x}^{{2r}}}={{x}^{0}}

\displaystyle {{x}^{{-10+r}}}\times {{x}^{{2r}}}={{x}^{0}}

\displaystyle {{x}^{{-10+r+2r}}}={{x}^{0}}

Extract the powers of x and find the value of r.

\displaystyle -10+r+2r=0

\displaystyle 3r-10=0

\displaystyle 3r=10

\displaystyle r=\frac{{10}}{3}

Since the value of r is a fraction, there is no term in the expansion the has the coefficient of x0 (independent of x).

Note: In any binomial expansion, the r value starts from 0 followed by 1,2,3… . When r=0, the term is 1, r=1, the term is 2, r=2 the term is 3 and so forth. We use the r value to find the position of a certain coefficient of x. But if that certain coefficient does not exist in the expansion, the r value will NOT be a whole number.


Small Group Tuition Additional Math (AMath) at Woodlands, Elementary Math (EMath) at Admiralty.