Mathematics

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


ANSWER

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


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