How do you write the partial fraction decomposition of the rational expression $\frac{x^{2}}{{(x + 1)}^{3}}$ $(x^2)/(x+1)^3$ ?

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How do you write the partial fraction decomposition of the rational expression $\frac{x^{2}}{{(x + 1)}^{3}}$ $(x^2)/(x+1)^3$ ?

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Answer 1 · 2017-03-29T19:21:03

$\frac{x^{2}}{{(x + 1)}^{3}} = \frac{1}{x + 1} - \frac{2}{{(x + 1)}^{2}} + \frac{1}{{(x + 1)}^{3}}$ $x^2/(x+1)^3 = 1/(x+1)-2/(x+1)^2+1/(x+1)^3$

Explanation:

$\frac{x^{2}}{{(x + 1)}^{3}} = \frac{A}{x + 1} + \frac{B}{{(x + 1)}^{2}} + \frac{C}{{(x + 1)}^{3}}$ $x^2/(x+1)^3 = A/(x+1)+B/(x+1)^2+C/(x+1)^3$

$\frac{x^{2}}{{(x + 1)}^{3}} = \frac{A {(x + 1)}^{2} + B (x + 1) + C}{{(x + 1)}^{3}}$ $color(white)(x^2/(x+1)^3) = (A(x+1)^2+B(x+1)+C)/(x+1)^3$

$\frac{x^{2}}{{(x + 1)}^{3}} = \frac{A (x^{2} + 2 x + 1) + B (x + 1) + C}{{(x + 1)}^{3}}$ $color(white)(x^2/(x+1)^3) = (A(x^2+2x+1)+B(x+1)+C)/(x+1)^3$

$\frac{x^{2}}{{(x + 1)}^{3}} = \frac{A x^{2} + (2 A + B) x + (A + B + C)}{{(x + 1)}^{3}}$ $color(white)(x^2/(x+1)^3) = (Ax^2+(2A+B)x+(A+B+C))/(x+1)^3$

Equating coefficients we have:

${ (A=1), (2A+B=0), (A+B+C=0) :}$

Hence:

${ (A=1), (B=-2), (C=1) :}$

So:

$x^2/(x+1)^3 = 1/(x+1)-2/(x+1)^2+1/(x+1)^3$

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How do you write the partial fraction decomposition of the rational expression $\frac{x^{2}}{{(x + 1)}^{3}}$ $(x^2)/(x+1)^3$ ?

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How do you write the partial fraction decomposition of the rational expression $\frac{x^{2}}{{(x + 1)}^{3}}$ $(x^2)/(x+1)^3$ ?