Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
-(z^2+1)/(z-(-1-sqrt(3)*i)/2)^2+2*z/(z-(-1-sqrt(3)*i)/2)
How do you -(z^2+1)/(z-(-1-sqrt(3)*i)/2)^2+2*z/(z-(-1-sqrt(3)*i)/2) in partial fractions?
The solution
You have entered
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2
- z - 1 2*z
------------------- + ----------------
2 ___
/ ___ \ -1 - \/ 3 *I
| -1 - \/ 3 *I| z - ------------
|z - ------------| 2
\ 2 /
$$\frac{2 z}{z - \frac{-1 - \sqrt{3} i}{2}} + \frac{- z^{2} - 1}{\left(z - \frac{-1 - \sqrt{3} i}{2}\right)^{2}}$$
(-z^2 - 1)/(z - (-1 - sqrt(3)*i)/2)^2 + (2*z)/(z - (-1 - sqrt(3)*i)/2)
Fraction decomposition
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1 - 2*(1 + i*sqrt(3))/(1 + 2*z + i*sqrt(3))^2
$$1 - \frac{2 \left(1 + \sqrt{3} i\right)}{\left(2 z + 1 + \sqrt{3} i\right)^{2}}$$
/ ___\
2*\1 + I*\/ 3 /
1 - --------------------
2
/ ___\
\1 + 2*z + I*\/ 3 /
General simplification
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/ 2 / ___\\
4*\-1 - z + z*\1 + 2*z + I*\/ 3 //
-----------------------------------
2
/ ___\
\1 + 2*z + I*\/ 3 /
$$\frac{4 \left(- z^{2} + z \left(2 z + 1 + \sqrt{3} i\right) - 1\right)}{\left(2 z + 1 + \sqrt{3} i\right)^{2}}$$
4*(-1 - z^2 + z*(1 + 2*z + i*sqrt(3)))/(1 + 2*z + i*sqrt(3))^2
Assemble expression
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2
-1 - z 2*z
------------------ + ---------------
2 ___
/ ___\ 1 I*\/ 3
|1 I*\/ 3 | - + z + -------
|- + z + -------| 2 2
\2 2 /
$$\frac{2 z}{z + \frac{1}{2} + \frac{\sqrt{3} i}{2}} + \frac{- z^{2} - 1}{\left(z + \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)^{2}}$$
(-1 - z^2)/(1/2 + z + i*sqrt(3)/2)^2 + 2*z/(1/2 + z + i*sqrt(3)/2)
Powers
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2
-1 - z 2*z
------------------ + ---------------
2 ___
/ ___\ 1 I*\/ 3
|1 I*\/ 3 | - + z + -------
|- + z + -------| 2 2
\2 2 /
$$\frac{2 z}{z + \frac{1}{2} + \frac{\sqrt{3} i}{2}} + \frac{- z^{2} - 1}{\left(z + \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)^{2}}$$
(-1 - z^2)/(1/2 + z + i*sqrt(3)/2)^2 + 2*z/(1/2 + z + i*sqrt(3)/2)
Rational denominator
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3 3 3 3 3 2 3 3
/ ___\ / ___\ 3 / ___\ 2 / ___\ ___ / ___\ / ___\ / ___\ ___ 2 / ___\
- 4*\1 + 2*z - I*\/ 3 / - 8*z*\1 + 2*z - I*\/ 3 / - 8*z *\1 + 2*z - I*\/ 3 / - 4*z *\1 + 2*z - I*\/ 3 / - 4*I*\/ 3 *\1 + 2*z - I*\/ 3 / + 4*z*\1 + 2*z + I*\/ 3 / *\1 + 2*z - I*\/ 3 / - 4*I*\/ 3 *z *\1 + 2*z - I*\/ 3 /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3
/ 2\
\4 + 4*z + 4*z /
$$\frac{- 8 z^{3} \left(2 z + 1 - \sqrt{3} i\right)^{3} - 4 z^{2} \left(2 z + 1 - \sqrt{3} i\right)^{3} - 4 \sqrt{3} i z^{2} \left(2 z + 1 - \sqrt{3} i\right)^{3} + 4 z \left(2 z + 1 - \sqrt{3} i\right)^{3} \left(2 z + 1 + \sqrt{3} i\right)^{2} - 8 z \left(2 z + 1 - \sqrt{3} i\right)^{3} - 4 \left(2 z + 1 - \sqrt{3} i\right)^{3} - 4 \sqrt{3} i \left(2 z + 1 - \sqrt{3} i\right)^{3}}{\left(4 z^{2} + 4 z + 4\right)^{3}}$$
(-4*(1 + 2*z - i*sqrt(3))^3 - 8*z*(1 + 2*z - i*sqrt(3))^3 - 8*z^3*(1 + 2*z - i*sqrt(3))^3 - 4*z^2*(1 + 2*z - i*sqrt(3))^3 - 4*i*sqrt(3)*(1 + 2*z - i*sqrt(3))^3 + 4*z*(1 + 2*z + i*sqrt(3))^2*(1 + 2*z - i*sqrt(3))^3 - 4*i*sqrt(3)*z^2*(1 + 2*z - i*sqrt(3))^3)/(4 + 4*z + 4*z^2)^3
Combinatorics
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/ 2 ___\
4*\-1 + z + z + I*z*\/ 3 /
---------------------------
2
/ ___\
\1 + 2*z + I*\/ 3 /
$$\frac{4 \left(z^{2} + z + \sqrt{3} i z - 1\right)}{\left(2 z + 1 + \sqrt{3} i\right)^{2}}$$
4*(-1 + z + z^2 + i*z*sqrt(3))/(1 + 2*z + i*sqrt(3))^2
Common denominator
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___
1 + I*\/ 3
1 - ---------------------------------------
2 ___ ___
-1 + 2*z + 2*z + I*\/ 3 + 2*I*z*\/ 3
$$1 - \frac{1 + \sqrt{3} i}{2 z^{2} + 2 z + 2 \sqrt{3} i z - 1 + \sqrt{3} i}$$
1 - (1 + i*sqrt(3))/(-1 + 2*z + 2*z^2 + i*sqrt(3) + 2*i*z*sqrt(3))
Trigonometric part
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2
-1 - z 2*z
------------------ + ---------------
2 ___
/ ___\ 1 I*\/ 3
|1 I*\/ 3 | - + z + -------
|- + z + -------| 2 2
\2 2 /
$$\frac{2 z}{z + \frac{1}{2} + \frac{\sqrt{3} i}{2}} + \frac{- z^{2} - 1}{\left(z + \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)^{2}}$$
(-1 - z^2)/(1/2 + z + i*sqrt(3)/2)^2 + 2*z/(1/2 + z + i*sqrt(3)/2)
Combining rational expressions
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/ 2 / ___\\
4*\-1 - z + z*\1 + 2*z + I*\/ 3 //
-----------------------------------
2
/ ___\
\1 + 2*z + I*\/ 3 /
$$\frac{4 \left(- z^{2} + z \left(2 z + 1 + \sqrt{3} i\right) - 1\right)}{\left(2 z + 1 + \sqrt{3} i\right)^{2}}$$
4*(-1 - z^2 + z*(1 + 2*z + i*sqrt(3)))/(1 + 2*z + i*sqrt(3))^2
Numerical answer
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(-1.0 - z^2)/(0.5 + z + 0.866025403784439*i)^2 + 2.0*z/(0.5 + z + 0.866025403784439*i)
(-1.0 - z^2)/(0.5 + z + 0.866025403784439*i)^2 + 2.0*z/(0.5 + z + 0.866025403784439*i)