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

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

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Answer 1 · 2016-02-01T08:55:31

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Explanation:

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

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

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

Equating the numerators

$x + 1 = A x (x - 2) + B (x - 2) + C x^{2}$ $x+1 = Ax(x-2)+B(x-2)+Cx^2$

To make this easy for us, let us start by taking $x = 0$ $x=0$ and plug in the value, we get.

$0 + 1 = A (0) (0 - 2) + B (0 - 2) + C {(0)}^{2}$ $0+1=A(0)(0-2)+B(0-2)+C(0)^2$
$1 = 0 - 2 B + 0$ $1=0-2B+0$

$- 2 B = 1$ $-2B=1$
$B = - \frac{1}{2}$ $B=-1/2$

Note $x = 0$ $x=0$ helped us eliminate $A$ $A$ and $C$ $C$ now let us take $x = 2$ $x=2$
$2 + 1 = A (2) (2 - 2) + B (2 - 2) + C (2^{2})$ $2+1 = A(2)(2-2)+B(2-2)+C(2^2)$
$3 = A (2) (0) + B (0) + 4 C$ $3=A(2)(0)+B(0)+4C$
$3 = 0 + 0 + 4 C$ $3=0+0+4C$

$4 C = 3$ $4C=3$
$C = \frac{3}{4}$ $C=3/4$

We have to find $A$ $A$ Let us plug in x=1
$1 + 1 = A (1) (1 - 2) + B (1 - 2) + C (1^{2})$ $1+1=A(1)(1-2)+B(1-2)+C(1^2)$
$2 = A (1) (- 1) - B + C$ $2=A(1)(-1)-B+C$

$- A - B + C = 2$ $-A-B+C = 2$
$- A - (- \frac{1}{2}) + \frac{3}{4} = 2$ $-A-(-1/2)+3/4=2$

$- A + \frac{1}{2} + \frac{3}{4} = 2$ $-A+1/2+3/4=2$

$- A + \frac{2}{4} + \frac{3}{4} = 2$ $-A+2/4+3/4=2$
$- A + \frac{5}{4} = 2$ $-A+5/4=2$

$- A = 2 - \frac{5}{4}$ $-A=2-5/4$

$- A = \frac{8 - 5}{4}$ $-A=(8-5)/4$

$- A = \frac{3}{4}$ $-A=3/4$

$A = - \frac{3}{4}$ $A=-3/4$

$\frac{x + 1}{x^{2} (x - 2)} = - \frac{3}{4 x} - \frac{1}{2 x^{2}} + \frac{3}{4 (x - 2)}$ $(x+1)/(x^2(x-2)) = -3/(4x)-1/(2x^2) + 3/(4(x-2))$

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

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