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Question Number 26733 by math solver last updated on 28/Dec/17

Commented by prakash jain last updated on 28/Dec/17

${(\frac{3}{2} x^{3} - \frac{1}{3 x})}^{9} =^{9} C_{i} {(\frac{3}{2})}^{i} x^{2 i} {(- 1)}^{9 - i} {(\frac{1}{3 x})}^{9 - i}$ $^{9} C_{i} {(\frac{3}{2})}^{i} {(- 1)}^{9 - i} {(\frac{1}{3})}^{9 - i} x^{4 i - 9}$ $c o e f f c i e n t o f c o n s t a n t t e r m$ $3 i - 9 = 0 \Rightarrow i = 3$ $^{9} C_{2} {(\frac{3}{2})}^{3} {(- 1)}^{9 - 3} {(\frac{1}{3})}^{9 - 3}$ $= \frac{9!}{3! 6!} \times \frac{3^{3}}{2^{3}} \times \frac{1}{3^{6}} = \frac{9 \times 8 \times 7}{1 \times 2 \times 3} \times \frac{1}{8} \times \frac{1}{3 \times 9} = \frac{7}{18}$ $3 i - 9 = - 3 \Rightarrow i = 2$ $2 \times \frac{9!}{2! 7!} \times \frac{3^{2}}{2^{2}} \times {(- 1)}^{9 - 2} \times \frac{1}{3^{7}} = \frac{9 \times 8}{4 \times 3^{5}} = - \frac{2}{27}$ $\frac{7}{18} - \frac{2}{27} = \frac{17}{54}$
	Answered by mrW1 last updated on 28/Dec/17

	$(1 + x + 2 x^{3}) {(\frac{3}{2} x^{2} - \frac{1}{3 x})}^{9}$ $= \frac{1}{x^{9}} (1 + x + 2 x^{3}) {(\frac{3}{2} x^{3} - \frac{1}{3})}^{9}$ $1 \times \frac{9 \times 8 \times 7}{3 \times 2 \times 1} \times {(\frac{3}{2})}^{3} {(- \frac{1}{3})}^{6} + 2 \times \frac{9 \times 8}{2 \times 1} \times {(\frac{3}{2})}^{2} {(- \frac{1}{3})}^{7}$ $= {(\frac{3}{2})}^{2} {(- \frac{1}{3})}^{6} [1 \times \frac{9 \times 8 \times 7}{3 \times 2 \times 1} \times (\frac{3}{2}) + 2 \times \frac{9 \times 8}{2 \times 1} \times (- \frac{1}{3})]$ $= {(\frac{3}{2})}^{2} {(- \frac{1}{3})}^{6} (126 - 24)$ $= \frac{1}{2^{2}} \times \frac{1}{3^{4}} \times 102$ $= \frac{17}{54}$ $\Rightarrow A n s w e r (A)$
	Commented by math solver last updated on 28/Dec/17

	$p l z e x p l a i n s t e p - 3 ?$ $h o w y o u f i n d t e r m i n d e p e n d e n t o f x ?$
	Commented by math solver last updated on 28/Dec/17

	$i t s s o m e t h i n g l i k e$ $9_{C_{3}} a n d 2 . 9_{C_{7}} .$
	Commented by mrW1 last updated on 28/Dec/17

	$x^{0} t e r m f r o m \frac{1}{x^{9}} (1 + x + 2 x^{3}) {(\frac{3}{2} x^{3} - \frac{1}{3})}^{9}$ $\equiv x^{9} t e r m f r o m (1 + x + 2 x^{3}) {(\frac{3}{2} x^{3} - \frac{1}{3})}^{9}$ $\equiv 1 \times x^{9} t e r m + x \times x^{8} t e r m + 2 x^{3} \times x^{6} t e r m$ $x^{9} t e r m o f {(\frac{3}{2} x^{3} - \frac{1}{3})}^{9} i s C_{3}^{9} \times {(\frac{3}{2} x^{3})}^{3} \times {(- \frac{1}{3})}^{6}$ $t h e r e i s n o x^{8} t e r m i n {(\frac{3}{2} x^{3} - \frac{1}{3})}^{9}$ $x^{6} t e r m o f {(\frac{3}{2} x^{3} - \frac{1}{3})}^{9} i s C_{2}^{9} \times {(\frac{3}{2} x^{3})}^{2} \times {(- \frac{1}{3})}^{7}$ $t h e r e f o r e t h e c o e f f i c i e n t o f x^{0} t e r m i s$ $1 \times C_{3}^{9} \times {(\frac{3}{2})}^{3} \times {(- \frac{1}{3})}^{6} + 2 \times C_{2}^{9} \times {(\frac{3}{2})}^{2} \times {(- \frac{1}{3})}^{7}$
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