Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
(6/(a-1)-10/(a-1)^2)/(10/(a^2-1)-(2*a+2)/(a-1))
How do you (6/(a-1)-10/(a-1)^2)/(10/(a^2-1)-(2*a+2)/(a-1)) in partial fractions?
The solution
You have entered
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6 10
----- - --------
a - 1 2
(a - 1)
----------------
10 2*a + 2
------ - -------
2 a - 1
a - 1
$$\frac{- \frac{10}{\left(a - 1\right)^{2}} + \frac{6}{a - 1}}{\frac{10}{a^{2} - 1} - \frac{2 a + 2}{a - 1}}$$
(6/(a - 1) - 10/(a - 1)^2)/(10/(a^2 - 1) - (2*a + 2)/(a - 1))
Fraction decomposition
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-10/(-1 + a) + (32 + 7*a)/(-4 + a^2 + 2*a)
$$\frac{7 a + 32}{a^{2} + 2 a - 4} - \frac{10}{a - 1}$$
10 32 + 7*a
- ------ + -------------
-1 + a 2
-4 + a + 2*a
General simplification
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2
8 - 3*a + 5*a
-----------------
2 3
4 + a + a - 6*a
$$\frac{- 3 a^{2} + 5 a + 8}{a^{3} + a^{2} - 6 a + 4}$$
(8 - 3*a^2 + 5*a)/(4 + a^2 + a^3 - 6*a)
Numerical answer
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(6.0/(-1.0 + a) - 10.0/(-1.0 + a)^2)/(10.0/(-1.0 + a^2) - (2.0 + 2.0*a)/(-1.0 + a))
(6.0/(-1.0 + a) - 10.0/(-1.0 + a)^2)/(10.0/(-1.0 + a^2) - (2.0 + 2.0*a)/(-1.0 + a))
Combining rational expressions
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/ 2\
\-1 + a /*(-8 + 3*a)
---------------------------------------
/ / 2\\
(-1 + a)*\-5 + 5*a - (1 + a)*\-1 + a //
$$\frac{\left(3 a - 8\right) \left(a^{2} - 1\right)}{\left(a - 1\right) \left(5 a - \left(a + 1\right) \left(a^{2} - 1\right) - 5\right)}$$
(-1 + a^2)*(-8 + 3*a)/((-1 + a)*(-5 + 5*a - (1 + a)*(-1 + a^2)))
Combinatorics
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-(1 + a)*(-8 + 3*a)
------------------------
/ 2 \
(-1 + a)*\-4 + a + 2*a/
$$- \frac{\left(a + 1\right) \left(3 a - 8\right)}{\left(a - 1\right) \left(a^{2} + 2 a - 4\right)}$$
-(1 + a)*(-8 + 3*a)/((-1 + a)*(-4 + a^2 + 2*a))
Assemble expression
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10 6
- --------- + ------
2 -1 + a
(-1 + a)
--------------------
10 2 + 2*a
------- - -------
2 -1 + a
-1 + a
$$\frac{\frac{6}{a - 1} - \frac{10}{\left(a - 1\right)^{2}}}{\frac{10}{a^{2} - 1} - \frac{2 a + 2}{a - 1}}$$
(-10/(-1 + a)^2 + 6/(-1 + a))/(10/(-1 + a^2) - (2 + 2*a)/(-1 + a))
Trigonometric part
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10 6
- --------- + ------
2 -1 + a
(-1 + a)
--------------------
10 2 + 2*a
------- - -------
2 -1 + a
-1 + a
$$\frac{\frac{6}{a - 1} - \frac{10}{\left(a - 1\right)^{2}}}{\frac{10}{a^{2} - 1} - \frac{2 a + 2}{a - 1}}$$
(-10/(-1 + a)^2 + 6/(-1 + a))/(10/(-1 + a^2) - (2 + 2*a)/(-1 + a))
Rational denominator
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2 2 3 2 2
10 - 10*a - 10*a + 6*(-1 + a) + 10*a - 6*a *(-1 + a)
--------------------------------------------------------
2 / 2 3\
(-1 + a) *\8 - 12*a + 2*a + 2*a /
$$\frac{10 a^{3} - 6 a^{2} \left(a - 1\right)^{2} - 10 a^{2} - 10 a + 6 \left(a - 1\right)^{2} + 10}{\left(a - 1\right)^{2} \left(2 a^{3} + 2 a^{2} - 12 a + 8\right)}$$
(10 - 10*a - 10*a^2 + 6*(-1 + a)^2 + 10*a^3 - 6*a^2*(-1 + a)^2)/((-1 + a)^2*(8 - 12*a + 2*a^2 + 2*a^3))
Common denominator
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/ 2\
-\-8 - 5*a + 3*a /
-------------------
2 3
4 + a + a - 6*a
$$- \frac{3 a^{2} - 5 a - 8}{a^{3} + a^{2} - 6 a + 4}$$
-(-8 - 5*a + 3*a^2)/(4 + a^2 + a^3 - 6*a)
Powers
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10 6
- --------- + ------
2 -1 + a
(-1 + a)
--------------------
10 2 + 2*a
------- - -------
2 -1 + a
-1 + a
$$\frac{\frac{6}{a - 1} - \frac{10}{\left(a - 1\right)^{2}}}{\frac{10}{a^{2} - 1} - \frac{2 a + 2}{a - 1}}$$
10 6
- --------- + ------
2 -1 + a
(-1 + a)
--------------------
10 -2 - 2*a
------- + --------
2 -1 + a
-1 + a
$$\frac{\frac{6}{a - 1} - \frac{10}{\left(a - 1\right)^{2}}}{\frac{- 2 a - 2}{a - 1} + \frac{10}{a^{2} - 1}}$$
(-10/(-1 + a)^2 + 6/(-1 + a))/(10/(-1 + a^2) + (-2 - 2*a)/(-1 + a))