The rational function is improper

MATHEMATICS · Single correct · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

The rational function 2x3−3x+4(x−2)(x2+1)\dfrac{2x^{3}-3x+4}{(x-2)(x^{2}+1)} is improper. After dividing, one obtains a proper fraction that needs to be decomposed. Which of the following is the correct general form for that proper fraction part? (a) A(x−2)2+Bx+Cx2+1\dfrac{A}{(x-2)^{2}}+\dfrac{Bx+C}{x^{2}+1} (b) Ax−2+Bx2+1\dfrac{A}{x-2}+\dfrac{B}{x^{2}+1} (c) Ax−2+Bx+Cx2+1\dfrac{A}{x-2}+\dfrac{Bx+C}{x^{2}+1} (d) Ax−2+Bx+C(x2+1)2\dfrac{A}{x-2}+\dfrac{Bx+C}{(x^{2}+1)^{2}}

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