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For Example, what is the coeffecint of $x^{-1}$ in the expansion of $$ (\frac{1}{2x}+3x)^5​(x+​1)^4\text{?} $$ How can I find the coefficient without expanding by hand?

Sebastian Garcia
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  • The way that I had Written the two binomials caused them to be misinterpreted so I have edited the post, sorry for the inconvenience. – Sebastian Garcia Oct 10 '18 at 19:43

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$x^{-1}$ can come from $1\cdot x^{-1}$, $x\cdot x^{-2}$, $x^2\cdot x^{-3}$, $x^3\cdot x^{-4}$, or $x^4\cdot x^{-5}$. The coefficient of $x^{-1}$ in the product is the sum of the product of the coefficients of these terms in each binomial. We have $$\sum_{i=0}^4 \frac{1}{32}\begin{pmatrix}4\\i\end{pmatrix}\begin{pmatrix}5\\i+1\end{pmatrix}=\frac{63}{16}$$

HackerBoss
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