Mathematics

Resolve into partial fractions $$\displaystyle \frac{x^2+2}{(x+1)^3(x-2)}$$


ANSWER

$$\displaystyle - \frac{6}{(x+2)}+\frac{6}{(x+1)}-\frac{5}{(x+1)^2}+\frac{3}{(x+1)^3}$$


SOLUTION
Let $$\displaystyle \frac { x^{ 2 }+2 }{ (x+1)^{ 3 }(x-2) } =\frac { A }{ \left( x+1 \right)  } +\frac { B }{ { \left( x+1 \right)  }^{ 2 } } +\frac { C }{ { \left( x+1 \right)  }^{ 3 } } +\frac { D }{ \left( x-2 \right)  } $$
$$\Rightarrow \left( { x }^{ 2 }+2 \right) =A{ \left( x+1 \right)  }^{ 2 }\left( x-2 \right) +B\left( x+1 \right) \left( x-2 \right) +C\left( x-2 \right) +D{ \left( x+1 \right)  }^{ 3 }$$
$$\Rightarrow { x }^{ 2 }+2=A\left( { x }^{ 3 }-x-2 \right) +B\left( { x }^{ 2 }-x-2 \right) +C\left( x-2 \right) +D\left( { x }^{ 3 }+3{ x }^{ 2 }+3x+1 \right) $$
On comparing we get
$$A+D=0,B+D=1,-A-B+C+3=0,-2A-2B-2C+D=2\\ \Rightarrow A=6,B=-5,C=3,D=-6$$
Hence
$$\displaystyle \frac { x^{ 2 }+2 }{ (x+1)^{ 3 }(x-2) } =\frac { 6 }{ \left( x+1 \right)  } -\frac { 5 }{ { \left( x+1 \right)  }^{ 2 } } +\frac { 3 }{ { \left( x+1 \right)  }^{ 3 } } -\frac { 6 }{ \left( x-2 \right)  } $$
Hence, option 'B' is correct.
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Single Correct Medium Published on 17th 09, 2020
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