For the rational function

MATHEMATICS · Single correct · AP · Calculus BC · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

For the rational function 5x2+1(x−2)3(x+1)\displaystyle \frac{5x^{2}+1}{(x-2)^{3}(x+1)}, which one of the following sets of terms must necessarily appear in its partial–fraction decomposition? (a) Ax−2\dfrac{A}{x-2} (b) Ax−2+B(x+1)\dfrac{A}{x-2}+\dfrac{B}{(x+1)} (c) Ax−2+B(x−2)2+C(x−2)3\dfrac{A}{x-2}+\dfrac{B}{(x-2)^2}+\dfrac{C}{(x-2)^3} (d) Ax−2+B(x−2)2\dfrac{A}{x-2}+\dfrac{B}{(x-2)^2}

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