boitshupoO

2021-05-14

Find the coefficient of${x}^{7}$ in the expression $\left({x}^{3}+\frac{4}{{x}^{2}}{\right)}^{4}$

Delorenzoz

Solution:
$\left({x}^{3}+\frac{4}{{x}^{2}}{\right)}^{4}=\left(\left({x}^{3}+\frac{4}{{x}^{2}}{\right)}^{2}\right)$
$=\left({x}^{6}+\frac{16}{{x}^{4}}+8x{\right)}^{2}$
$={x}^{12}+\frac{256}{{x}^{8}}+64{x}^{2}+2\left[{x}^{6}\cdot \frac{16}{{x}^{4}}+{x}^{6}\cdot 8x+\frac{16}{{x}^{4}}\cdot 8x\right]$
So: $\left(a+b+c{\right)}^{2}={a}^{2}+{b}^{2}+{c}^{2}+2\left(ab+bc+ac\right)$
Required form =8x
Coefficient ${x}^{2}=8$

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