Find coefficient of x3 in the expansion of x2 - 3-2-x 9

Formula → Binomial expansion, (a+b)^n = ^nC_0 a^n b^0+ ^nC_1 a^n 1 b^1 +.... ^nC_n a^o b^n =∑ _r=0^n ^nC_r a^n r b^r In Question → (x ...

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Byju's Answer

Standard XII

Mathematics

Binomial Expression

Find coeffici...

Question

Find coefficient of $x^{3}$ in the expansion of ${(x^{2} + \frac{3 \sqrt{2}}{x})}^{9}$

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Solution

$F o r m u l a \to$
Binomial expansion, $(a + b)^{n} =^{n} C_{0} a^{n} b^{0} +^{n} C_{1} a^{n - 1} b^{1} + . . . .^{n} C_{n} a^{o} b^{n}$
$= n \sum r = 0^{n} C_{r} a^{n - r} b^{r}$
In Question $\to$
${(x^{2} + \frac{3 \sqrt{2}}{x})}^{9} = 9 \sum r = 0^{9} C_{r} (x^{2})^{9 - r} {(\frac{3 \sqrt{2}}{x})}^{r}$ [using Binomial expansion formula Taking $n = 9$ ]
$= 9 \sum r = 0^{9} C_{r} (x)^{18 - 2 r - r} (3 \sqrt{2})^{r}$
$== 9 \sum r = 0^{9} C_{r} (x)^{18 - 3 r} (2 \sqrt{2})^{r}$
$\Rightarrow$ Here power of $x$ is $18 - 3 r 18 - 3 r = 3$ when $r = 5$
on Solving $[18 - 3 = 3 r 5 = r]$
$∴$ Coefficient of $x^{3} =^{9} C_{r} (3 \sqrt{2})^{5}$
$= \frac{9!}{5! (9 - 5)!} (3 \sqrt{2})^{5} = \frac{9!}{5! 4!} (3)^{5} (\sqrt{2})^{5}$

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Find coefficient of x3 in the expansion of (x2+3√2x)9

Find coefficient of $x^{3}$ in the expansion of ${(x^{2} + \frac{3 \sqrt{2}}{x})}^{9}$