Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
(z-2)/(6*z+(z-2)*(z-2))+(6/(z*z*z-8))
How do you (z-2)/(6*z+(z-2)*(z-2))+(6/(z*z*z-8)) in partial fractions?
The solution
You have entered
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z - 2 6 --------------------- + --------- 6*z + (z - 2)*(z - 2) z*z*z - 8
$$\frac{z - 2}{6 z + \left(z - 2\right) \left(z - 2\right)} + \frac{6}{z z z - 8}$$
(z - 2)/(6*z + (z - 2)*(z - 2)) + 6/((z*z)*z - 8)
Fraction decomposition
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1/(2*(-2 + z)) + (-8 + z)/(2*(4 + z^2 + 2*z))
$$\frac{z - 8}{2 \left(z^{2} + 2 z + 4\right)} + \frac{1}{2 \left(z - 2\right)}$$
1 -8 + z
---------- + ----------------
2*(-2 + z) / 2 \
2*\4 + z + 2*z/
General simplification
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2
10 + z - 4*z
-------------
3
-8 + z
$$\frac{z^{2} - 4 z + 10}{z^{3} - 8}$$
(10 + z^2 - 4*z)/(-8 + z^3)
Common denominator
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2
10 + z - 4*z
-------------
3
-8 + z
$$\frac{z^{2} - 4 z + 10}{z^{3} - 8}$$
(10 + z^2 - 4*z)/(-8 + z^3)
Combinatorics
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2
10 + z - 4*z
-----------------------
/ 2 \
(-2 + z)*\4 + z + 2*z/
$$\frac{z^{2} - 4 z + 10}{\left(z - 2\right) \left(z^{2} + 2 z + 4\right)}$$
(10 + z^2 - 4*z)/((-2 + z)*(4 + z^2 + 2*z))
Powers
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6 -2 + z
------- + ---------------
3 2
-8 + z (-2 + z) + 6*z
$$\frac{z - 2}{6 z + \left(z - 2\right)^{2}} + \frac{6}{z^{3} - 8}$$
6/(-8 + z^3) + (-2 + z)/((-2 + z)^2 + 6*z)
Numerical answer
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6.0/(-8.0 + z^3) + (-2.0 + z)/(4.0*(-1 + 0.5*z)^2 + 6.0*z)
6.0/(-8.0 + z^3) + (-2.0 + z)/(4.0*(-1 + 0.5*z)^2 + 6.0*z)
Combining rational expressions
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2 / 3\
6*(-2 + z) + 36*z + \-8 + z /*(-2 + z)
---------------------------------------
/ 3\ / 2 \
\-8 + z /*\(-2 + z) + 6*z/
$$\frac{36 z + 6 \left(z - 2\right)^{2} + \left(z - 2\right) \left(z^{3} - 8\right)}{\left(6 z + \left(z - 2\right)^{2}\right) \left(z^{3} - 8\right)}$$
(6*(-2 + z)^2 + 36*z + (-8 + z^3)*(-2 + z))/((-8 + z^3)*((-2 + z)^2 + 6*z))
Rational denominator
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4 3 2
16 + z - 2*z + 6*(-2 + z) + 28*z
-----------------------------------
/ 3\ / 2 \
\-8 + z /*\4 + z + 2*z/
$$\frac{z^{4} - 2 z^{3} + 28 z + 6 \left(z - 2\right)^{2} + 16}{\left(z^{3} - 8\right) \left(z^{2} + 2 z + 4\right)}$$
(16 + z^4 - 2*z^3 + 6*(-2 + z)^2 + 28*z)/((-8 + z^3)*(4 + z^2 + 2*z))
Assemble expression
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6 -2 + z
------- + ---------------
3 2
-8 + z (-2 + z) + 6*z
$$\frac{z - 2}{6 z + \left(z - 2\right)^{2}} + \frac{6}{z^{3} - 8}$$
6/(-8 + z^3) + (-2 + z)/((-2 + z)^2 + 6*z)
Trigonometric part
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6 -2 + z
------- + ---------------
3 2
-8 + z (-2 + z) + 6*z
$$\frac{z - 2}{6 z + \left(z - 2\right)^{2}} + \frac{6}{z^{3} - 8}$$
6/(-8 + z^3) + (-2 + z)/((-2 + z)^2 + 6*z)