Mister Exam
Expression simplification/
Least common denominator/
(z*(z^2+6*z+1)-(4*z*(z+1)))/(z-1)^3+z*(z-1)/(z+1)^3
Least common denominator (z*(z^2+6*z+1)-(4*z*(z+1)))/(z-1)^3+z*(z-1)/(z+1)^3
The solution
You have entered
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/ 2 \
z*\z + 6*z + 1/ - 4*z*(z + 1) z*(z - 1)
------------------------------ + ---------
3 3
(z - 1) (z + 1)
$$\frac{z \left(z - 1\right)}{\left(z + 1\right)^{3}} + \frac{z \left(\left(z^{2} + 6 z\right) + 1\right) - 4 z \left(z + 1\right)}{\left(z - 1\right)^{3}}$$
(z*(z^2 + 6*z + 1) - 4*z*(z + 1))/(z - 1)^3 + (z*(z - 1))/(z + 1)^3
Fraction decomposition
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1 + 1/(1 + z) - 3/(1 + z)^2 + 2/(1 + z)^3 + 4/(-1 + z)^2 + 5/(-1 + z)
$$1 + \frac{1}{z + 1} - \frac{3}{\left(z + 1\right)^{2}} + \frac{2}{\left(z + 1\right)^{3}} + \frac{5}{z - 1} + \frac{4}{\left(z - 1\right)^{2}}$$
1 3 2 4 5
1 + ----- - -------- + -------- + --------- + ------
1 + z 2 3 2 -1 + z
(1 + z) (1 + z) (-1 + z)
General simplification
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/ 4 3 / 2 \\
z*\(-1 + z) + (1 + z) *\-3 + z + 2*z//
----------------------------------------
3 3
(1 + z) *(-1 + z)
$$\frac{z \left(\left(z - 1\right)^{4} + \left(z + 1\right)^{3} \left(z^{2} + 2 z - 3\right)\right)}{\left(z - 1\right)^{3} \left(z + 1\right)^{3}}$$
z*((-1 + z)^4 + (1 + z)^3*(-3 + z^2 + 2*z))/((1 + z)^3*(-1 + z)^3)
Combinatorics
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/ 2 \ / 2\
z*\1 + z + 6*z/*\2 + z + z /
-----------------------------
3 2
(1 + z) *(-1 + z)
$$\frac{z \left(z^{2} + z + 2\right) \left(z^{2} + 6 z + 1\right)}{\left(z - 1\right)^{2} \left(z + 1\right)^{3}}$$
z*(1 + z^2 + 6*z)*(2 + z + z^2)/((1 + z)^3*(-1 + z)^2)
Combining rational expressions
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/ 4 3 \
z*\(-1 + z) + (1 + z) *(-3 - 4*z + z*(6 + z))/
-----------------------------------------------
3 3
(1 + z) *(-1 + z)
$$\frac{z \left(\left(z - 1\right)^{4} + \left(z + 1\right)^{3} \left(z \left(z + 6\right) - 4 z - 3\right)\right)}{\left(z - 1\right)^{3} \left(z + 1\right)^{3}}$$
z*((-1 + z)^4 + (1 + z)^3*(-3 - 4*z + z*(6 + z)))/((1 + z)^3*(-1 + z)^3)
Trigonometric part
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/ 2 \
z*\1 + z + 6*z/ - 4*z*(1 + z) z*(-1 + z)
------------------------------ + ----------
3 3
(-1 + z) (1 + z)
$$\frac{z \left(z - 1\right)}{\left(z + 1\right)^{3}} + \frac{- 4 z \left(z + 1\right) + z \left(z^{2} + 6 z + 1\right)}{\left(z - 1\right)^{3}}$$
(z*(1 + z^2 + 6*z) - 4*z*(1 + z))/(-1 + z)^3 + z*(-1 + z)/(1 + z)^3
Common denominator
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4 3 2
-1 + z + 6*z + 11*z + 15*z
1 + -----------------------------
4 5 2 3
1 + z + z + z - 2*z - 2*z
$$\frac{6 z^{4} + 11 z^{3} + 15 z^{2} + z - 1}{z^{5} + z^{4} - 2 z^{3} - 2 z^{2} + z + 1} + 1$$
1 + (-1 + z + 6*z^4 + 11*z^3 + 15*z^2)/(1 + z + z^4 + z^5 - 2*z^2 - 2*z^3)
Rational denominator
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4 3 / / 2 \ \
z*(-1 + z) + (1 + z) *\z*\1 + z + 6*z/ - 4*z*(1 + z)/
-------------------------------------------------------
3 3
(1 + z) *(-1 + z)
$$\frac{z \left(z - 1\right)^{4} + \left(z + 1\right)^{3} \left(- 4 z \left(z + 1\right) + z \left(z^{2} + 6 z + 1\right)\right)}{\left(z - 1\right)^{3} \left(z + 1\right)^{3}}$$
(z*(-1 + z)^4 + (1 + z)^3*(z*(1 + z^2 + 6*z) - 4*z*(1 + z)))/((1 + z)^3*(-1 + z)^3)
Assemble expression
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/ 2 \
z*\1 + z + 6*z/ - 4*z*(1 + z) z*(-1 + z)
------------------------------ + ----------
3 3
(-1 + z) (1 + z)
$$\frac{z \left(z - 1\right)}{\left(z + 1\right)^{3}} + \frac{- 4 z \left(z + 1\right) + z \left(z^{2} + 6 z + 1\right)}{\left(z - 1\right)^{3}}$$
(z*(1 + z^2 + 6*z) - 4*z*(1 + z))/(-1 + z)^3 + z*(-1 + z)/(1 + z)^3
Numerical answer
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(z*(1.0 + z^2 + 6.0*z) - 4.0*z*(1.0 + z))/(-1.0 + z)^3 + z*(-1.0 + z)/(1.0 + z)^3
(z*(1.0 + z^2 + 6.0*z) - 4.0*z*(1.0 + z))/(-1.0 + z)^3 + z*(-1.0 + z)/(1.0 + z)^3
Powers
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/ 2 \
z*\1 + z + 6*z/ - 4*z*(1 + z) z*(-1 + z)
------------------------------ + ----------
3 3
(-1 + z) (1 + z)
$$\frac{z \left(z - 1\right)}{\left(z + 1\right)^{3}} + \frac{- 4 z \left(z + 1\right) + z \left(z^{2} + 6 z + 1\right)}{\left(z - 1\right)^{3}}$$
(z*(1 + z^2 + 6*z) - 4*z*(1 + z))/(-1 + z)^3 + z*(-1 + z)/(1 + z)^3