Evaluate
\frac{x^{4}-2x^{3}-1}{x^{2}\left(x^{2}+1\right)}
Expand
\frac{x^{4}-2x^{3}-1}{x^{2}\left(x^{2}+1\right)}
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\frac{\left(x^{2}-1\right)\left(x^{2}+1\right)}{x^{2}\left(x^{2}+1\right)}-\frac{2xx^{2}}{x^{2}\left(x^{2}+1\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and x^{2}+1 is x^{2}\left(x^{2}+1\right). Multiply \frac{x^{2}-1}{x^{2}} times \frac{x^{2}+1}{x^{2}+1}. Multiply \frac{2x}{x^{2}+1} times \frac{x^{2}}{x^{2}}.
\frac{\left(x^{2}-1\right)\left(x^{2}+1\right)-2xx^{2}}{x^{2}\left(x^{2}+1\right)}
Since \frac{\left(x^{2}-1\right)\left(x^{2}+1\right)}{x^{2}\left(x^{2}+1\right)} and \frac{2xx^{2}}{x^{2}\left(x^{2}+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{x^{4}+x^{2}-x^{2}-1-2x^{3}}{x^{2}\left(x^{2}+1\right)}
Do the multiplications in \left(x^{2}-1\right)\left(x^{2}+1\right)-2xx^{2}.
\frac{x^{4}-1-2x^{3}}{x^{2}\left(x^{2}+1\right)}
Combine like terms in x^{4}+x^{2}-x^{2}-1-2x^{3}.
\frac{x^{4}-1-2x^{3}}{x^{4}+x^{2}}
Expand x^{2}\left(x^{2}+1\right).
\frac{\left(x^{2}-1\right)\left(x^{2}+1\right)}{x^{2}\left(x^{2}+1\right)}-\frac{2xx^{2}}{x^{2}\left(x^{2}+1\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and x^{2}+1 is x^{2}\left(x^{2}+1\right). Multiply \frac{x^{2}-1}{x^{2}} times \frac{x^{2}+1}{x^{2}+1}. Multiply \frac{2x}{x^{2}+1} times \frac{x^{2}}{x^{2}}.
\frac{\left(x^{2}-1\right)\left(x^{2}+1\right)-2xx^{2}}{x^{2}\left(x^{2}+1\right)}
Since \frac{\left(x^{2}-1\right)\left(x^{2}+1\right)}{x^{2}\left(x^{2}+1\right)} and \frac{2xx^{2}}{x^{2}\left(x^{2}+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{x^{4}+x^{2}-x^{2}-1-2x^{3}}{x^{2}\left(x^{2}+1\right)}
Do the multiplications in \left(x^{2}-1\right)\left(x^{2}+1\right)-2xx^{2}.
\frac{x^{4}-1-2x^{3}}{x^{2}\left(x^{2}+1\right)}
Combine like terms in x^{4}+x^{2}-x^{2}-1-2x^{3}.
\frac{x^{4}-1-2x^{3}}{x^{4}+x^{2}}
Expand x^{2}\left(x^{2}+1\right).
Examples
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Matrix
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Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
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