Mister Exam
How do you (z*z-1/2*z)/(z*z-z+1)*z/(z-1/2) in partial fractions?
The solution
You have entered
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z
z*z - -
2
-----------*z
z*z - z + 1
-------------
z - 1/2
$$\frac{z \frac{- \frac{z}{2} + z z}{\left(- z + z z\right) + 1}}{z - \frac{1}{2}}$$
(((z*z - z/2)/(z*z - z + 1))*z)/(z - 1/2)
General simplification
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2
z
----------
2
1 + z - z
$$\frac{z^{2}}{z^{2} - z + 1}$$
z^2/(1 + z^2 - z)
Fraction decomposition
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1 + (-1 + z)/(1 + z^2 - z)
$$\frac{z - 1}{z^{2} - z + 1} + 1$$
-1 + z
1 + ----------
2
1 + z - z
Trigonometric part
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/ 2 z\
z*|z - -|
\ 2/
-----------------------
/ 2 \
(-1/2 + z)*\1 + z - z/
$$\frac{z \left(z^{2} - \frac{z}{2}\right)}{\left(z - \frac{1}{2}\right) \left(z^{2} - z + 1\right)}$$
z*(z^2 - z/2)/((-1/2 + z)*(1 + z^2 - z))
Common denominator
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-1 + z
1 + ----------
2
1 + z - z
$$\frac{z - 1}{z^{2} - z + 1} + 1$$
1 + (-1 + z)/(1 + z^2 - z)
Numerical answer
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z*(z^2 - 0.5*z)/((-0.5 + z)*(1.0 + z^2 - z))
z*(z^2 - 0.5*z)/((-0.5 + z)*(1.0 + z^2 - z))
Combining rational expressions
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2
z
--------------
1 + z*(-1 + z)
$$\frac{z^{2}}{z \left(z - 1\right) + 1}$$
z^2/(1 + z*(-1 + z))
Powers
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/ 2 z\
z*|z - -|
\ 2/
-----------------------
/ 2 \
(-1/2 + z)*\1 + z - z/
$$\frac{z \left(z^{2} - \frac{z}{2}\right)}{\left(z - \frac{1}{2}\right) \left(z^{2} - z + 1\right)}$$
z*(z^2 - z/2)/((-1/2 + z)*(1 + z^2 - z))
Rational denominator
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/ 2\
2*z*\-z + 2*z /
---------------------------
/ 2\
(-1 + 2*z)*\2 - 2*z + 2*z /
$$\frac{2 z \left(2 z^{2} - z\right)}{\left(2 z - 1\right) \left(2 z^{2} - 2 z + 2\right)}$$
2*z*(-z + 2*z^2)/((-1 + 2*z)*(2 - 2*z + 2*z^2))
Assemble expression
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/ 2 z\
z*|z - -|
\ 2/
-----------------------
/ 2 \
(-1/2 + z)*\1 + z - z/
$$\frac{z \left(z^{2} - \frac{z}{2}\right)}{\left(z - \frac{1}{2}\right) \left(z^{2} - z + 1\right)}$$
z*(z^2 - z/2)/((-1/2 + z)*(1 + z^2 - z))