Question: Simplify: 1 + \dfrac{1}{x-1} +\dfrac{2x}{x²+1} -\dfrac{x}{x+1} +\dfrac{4x³}{x⁴+1}


Solution:
Given,

= 1 + \dfrac{1}{x-1} +\dfrac{2x}{x²+1} -\dfrac{x}{x+1} +\dfrac{4x³}{x⁴+1}

= \dfrac{x-1 +1}{x-1} +\dfrac{2x}{x²+1} -\dfrac{x}{x+1} +\dfrac{4x³}{x⁴+1}

= \dfrac{x}{x-1} +\dfrac{2x}{x²+1} -\dfrac{x}{x+1} +\dfrac{4x³}{x⁴+1}

= \dfrac{x}{x-1} -\dfrac{x}{x+1} +\dfrac{2x}{x²+1} +\dfrac{4x³}{x⁴+1}

= \dfrac{x(x+1) -x(x-1)}{(x-1)(x+1)} +\dfrac{2x}{x²+1} +\dfrac{4x³}{x⁴+1}

= \dfrac{x²+x-x²+x}{x²-1} +\dfrac{2x}{x²+1} +\dfrac{4x³}{x⁴+1}

= \dfrac{2x}{x²-1} +\dfrac{2x}{x²+1} +\dfrac{4x³}{x⁴+1}

= \dfrac{2x(x²+1) +2x(x²-1)}{(x²-1)(x²+1)} +\dfrac{4x³}{x⁴+1}

= \dfrac{2x(x²+1+x²-1)}{x⁴-1} +\dfrac{4x³}{x⁴+1}

= \dfrac{2x(2x²)}{x⁴-1} +\dfrac{4x³}{x⁴+1}

= \dfrac{4x³}{x⁴-1} +\dfrac{4x³}{x⁴+1}

= \dfrac{4x³(x⁴+1) +4x³(x⁴-1)}{(x⁴-1)(x⁴+1)}

= \dfrac{4x³(x⁴+1+x⁴-1)}{x⁸-1}

= \dfrac{4x³(2x⁴)}{x⁸-1}

= \dfrac{8x⁷}{x⁸-1}
= Answer

Related Notes and Solutions:

Here is the Website link to the guide of Simplification of Rational Expressions.

Here is the Page link to all the solutions of Simplification of Rational Expressions.

#SciPiPupil
#Simplification
#Algebra