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 · 2016-03-03T07:37:24

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

$x^{2} = A {(x + 1)}^{2} + B (x + 1) + C \to x^{2} = A (x^{2} + 2 x + 1) + B x + B + C = A x^{2} + 2 A x + A$ $x^2=A(x+1)^2+B(x+1)+C->x^2=A(x^2+2x+1)+Bx+B+C=Ax^2+2Ax+A$

$1 = A, 2 A + B = 0, A + B + C = 0 \to A = 1, B = - 2, C = 1$ $1=A,2A+B=0,A+B+C=0->A=1,B=-2,C=1$

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

Explanation:

This is a repeating factor so you resolve it into three partial fractions. Next multiply everything by the common denominator. Then equate the coefficients of the like terms on both sides and use substitution method or linear combination to solve for A,B,C.

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