How do you write the partial fraction decomposition of the rational expression $\frac{x - 5}{{(x - 2)}^{2}}$ $(x-5)/(x-2)^2$ ?

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Answer 1 · 2015-12-14T21:10:40

$\frac{x - 5}{{(x - 2)}^{2}} = \frac{1}{x - 2} - \frac{3}{{(x - 2)}^{2}}$ $(x-5)/(x-2)^2=1/(x-2)-3/(x-2)^2$

$\frac{x - 5}{{(x - 2)}^{2}} = \frac{A}{x - 2} + \frac{B}{{(x - 2)}^{2}}$ $(x-5)/(x-2)^2=A/(x-2)+B/(x-2)^2$

$x - 5 = A (x - 2) + B$ $x-5=A(x-2)+B$

$x - 5 = A x - 2 A + B$ $x-5=Ax-2A+B$

$x - 5 = x (A) + 1 (- 2 A + B)$ $x-5=x(A)+1(-2A+B)$

Thus, ${(A=1),(-2A+B=-5):}$

So, ${(A=1),(B=-3):}$

Plug back in:

$(x-5)/(x-2)^2=1/(x-2)-3/(x-2)^2$

How do you write the partial fraction decomposition of the rational expression (x-5)/(x-2)^2x−5(x−2)2(x-5)/(x-2)^2?