Mister Exam
How do you (z+1)/(z+2)^3/(z-1) in partial fractions?
The solution
You have entered
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/ z + 1 \ |--------| | 3| \(z + 2) / ---------- z - 1
$$\frac{\left(z + 1\right) \frac{1}{\left(z + 2\right)^{3}}}{z - 1}$$
((z + 1)/(z + 2)^3)/(z - 1)
General simplification
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1 + z
-----------------
3
(-1 + z)*(2 + z)
$$\frac{z + 1}{\left(z - 1\right) \left(z + 2\right)^{3}}$$
(1 + z)/((-1 + z)*(2 + z)^3)
Fraction decomposition
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-2/(9*(2 + z)^2) - 2/(27*(2 + z)) + 1/(3*(2 + z)^3) + 2/(27*(-1 + z))
$$- \frac{2}{27 \left(z + 2\right)} - \frac{2}{9 \left(z + 2\right)^{2}} + \frac{1}{3 \left(z + 2\right)^{3}} + \frac{2}{27 \left(z - 1\right)}$$
2 2 1 2
- ---------- - ---------- + ---------- + -----------
2 27*(2 + z) 3 27*(-1 + z)
9*(2 + z) 3*(2 + z)
Assemble expression
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1 + z
-----------------
3
(-1 + z)*(2 + z)
$$\frac{z + 1}{\left(z - 1\right) \left(z + 2\right)^{3}}$$
(1 + z)/((-1 + z)*(2 + z)^3)
Combinatorics
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1 + z
-----------------
3
(-1 + z)*(2 + z)
$$\frac{z + 1}{\left(z - 1\right) \left(z + 2\right)^{3}}$$
(1 + z)/((-1 + z)*(2 + z)^3)
Numerical answer
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0.125*(1.0 + z)/((1 + 0.5*z)^3*(-1.0 + z))
0.125*(1.0 + z)/((1 + 0.5*z)^3*(-1.0 + z))
Combining rational expressions
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1 + z
-----------------
3
(-1 + z)*(2 + z)
$$\frac{z + 1}{\left(z - 1\right) \left(z + 2\right)^{3}}$$
(1 + z)/((-1 + z)*(2 + z)^3)
Trigonometric part
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1 + z
-----------------
3
(-1 + z)*(2 + z)
$$\frac{z + 1}{\left(z - 1\right) \left(z + 2\right)^{3}}$$
(1 + z)/((-1 + z)*(2 + z)^3)
Rational denominator
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1 + z
-----------------
3
(-1 + z)*(2 + z)
$$\frac{z + 1}{\left(z - 1\right) \left(z + 2\right)^{3}}$$
(1 + z)/((-1 + z)*(2 + z)^3)
Powers
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1 + z
-----------------
3
(-1 + z)*(2 + z)
$$\frac{z + 1}{\left(z - 1\right) \left(z + 2\right)^{3}}$$
(1 + z)/((-1 + z)*(2 + z)^3)
Common denominator
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1 + z
---------------------------
4 3 2
-8 + z - 4*z + 5*z + 6*z
$$\frac{z + 1}{z^{4} + 5 z^{3} + 6 z^{2} - 4 z - 8}$$
(1 + z)/(-8 + z^4 - 4*z + 5*z^3 + 6*z^2)