Evaluate
\frac{5+2z-z^{2}}{\left(z+1\right)\left(z+2\right)}
Expand
\frac{5+2z-z^{2}}{\left(z+1\right)\left(z+2\right)}
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\frac{2\left(z+2\right)}{\left(z+1\right)\left(z+2\right)}-\frac{\left(z-1\right)\left(z+1\right)}{\left(z+1\right)\left(z+2\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of z+1 and z+2 is \left(z+1\right)\left(z+2\right). Multiply \frac{2}{z+1} times \frac{z+2}{z+2}. Multiply \frac{z-1}{z+2} times \frac{z+1}{z+1}.
\frac{2\left(z+2\right)-\left(z-1\right)\left(z+1\right)}{\left(z+1\right)\left(z+2\right)}
Since \frac{2\left(z+2\right)}{\left(z+1\right)\left(z+2\right)} and \frac{\left(z-1\right)\left(z+1\right)}{\left(z+1\right)\left(z+2\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{2z+4-z^{2}-z+z+1}{\left(z+1\right)\left(z+2\right)}
Do the multiplications in 2\left(z+2\right)-\left(z-1\right)\left(z+1\right).
\frac{2z+5-z^{2}}{\left(z+1\right)\left(z+2\right)}
Combine like terms in 2z+4-z^{2}-z+z+1.
\frac{2z+5-z^{2}}{z^{2}+3z+2}
Expand \left(z+1\right)\left(z+2\right).
\frac{2\left(z+2\right)}{\left(z+1\right)\left(z+2\right)}-\frac{\left(z-1\right)\left(z+1\right)}{\left(z+1\right)\left(z+2\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of z+1 and z+2 is \left(z+1\right)\left(z+2\right). Multiply \frac{2}{z+1} times \frac{z+2}{z+2}. Multiply \frac{z-1}{z+2} times \frac{z+1}{z+1}.
\frac{2\left(z+2\right)-\left(z-1\right)\left(z+1\right)}{\left(z+1\right)\left(z+2\right)}
Since \frac{2\left(z+2\right)}{\left(z+1\right)\left(z+2\right)} and \frac{\left(z-1\right)\left(z+1\right)}{\left(z+1\right)\left(z+2\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{2z+4-z^{2}-z+z+1}{\left(z+1\right)\left(z+2\right)}
Do the multiplications in 2\left(z+2\right)-\left(z-1\right)\left(z+1\right).
\frac{2z+5-z^{2}}{\left(z+1\right)\left(z+2\right)}
Combine like terms in 2z+4-z^{2}-z+z+1.
\frac{2z+5-z^{2}}{z^{2}+3z+2}
Expand \left(z+1\right)\left(z+2\right).
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
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
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}