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  4. If (x+1/x4(x+2))=(A/x)+(B/x2)+(C/x3)+(D/x4)+(E/x+2) then B+D+E is equal to

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Q. If $\frac{x+1}{x^{4}(x+2)}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x^{3}}+\frac{D}{x^{4}}+\frac{E}{x+2}$ then $B+D+E$ is equal to

TS EAMCET 2016

Solution:

Given,
$\frac{x+1}{x^{4}(x+2)}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x^{3}}+\frac{D}{x^{4}}+\frac{E}{x+2}$
$\Rightarrow x+1=A x^{3}(x+2)+B x^{2}(x+2)+C x(x+2) +D(x+2)+E x^{4}$
At $x=-1$,
$\Rightarrow -1+1=-A+B-C+D+E$
$\Rightarrow 0=B+D+E-A-C$
$\therefore B+D+E=A+C$