Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
(s^2-3s+9)/(9s^2-1)•(3s^2+s)/(s*3+27)
How do you (s^2-3s+9)/(9s^2-1)•(3s^2+s)/(s*3+27) in partial fractions?
The solution
You have entered
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2
s - 3*s + 9 / 2 \
------------*\3*s + s/
2
9*s - 1
-----------------------
s*3 + 27
$$\frac{\frac{\left(s^{2} - 3 s\right) + 9}{9 s^{2} - 1} \left(3 s^{2} + s\right)}{3 s + 27}$$
(((s^2 - 3*s + 9)/(9*s^2 - 1))*(3*s^2 + s))/(s*3 + 27)
Fraction decomposition
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-35/27 + s/9 + 73/(756*(-1 + 3*s)) + 351/(28*(9 + s))
$$\frac{s}{9} - \frac{35}{27} + \frac{73}{756 \left(3 s - 1\right)} + \frac{351}{28 \left(s + 9\right)}$$
35 s 73 351 - -- + - + -------------- + ---------- 27 9 756*(-1 + 3*s) 28*(9 + s)
General simplification
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/ 2 \ s*\9 + s - 3*s/ -------------------- / 2 \ 3*\-9 + 3*s + 26*s/
$$\frac{s \left(s^{2} - 3 s + 9\right)}{3 \left(3 s^{2} + 26 s - 9\right)}$$
s*(9 + s^2 - 3*s)/(3*(-9 + 3*s^2 + 26*s))
Trigonometric part
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/ 2\ / 2 \ \s + 3*s /*\9 + s - 3*s/ ------------------------- / 2\ \-1 + 9*s /*(27 + 3*s)
$$\frac{\left(3 s^{2} + s\right) \left(s^{2} - 3 s + 9\right)}{\left(3 s + 27\right) \left(9 s^{2} - 1\right)}$$
(s + 3*s^2)*(9 + s^2 - 3*s)/((-1 + 9*s^2)*(27 + 3*s))
Assemble expression
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/ 2\ / 2 \ \s + 3*s /*\9 + s - 3*s/ ------------------------- / 2\ \-1 + 9*s /*(27 + 3*s)
$$\frac{\left(3 s^{2} + s\right) \left(s^{2} - 3 s + 9\right)}{\left(3 s + 27\right) \left(9 s^{2} - 1\right)}$$
(s + 3*s^2)*(9 + s^2 - 3*s)/((-1 + 9*s^2)*(27 + 3*s))
Rational denominator
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/ 2\ / 2 \ \s + 3*s /*\9 + s - 3*s/ ------------------------- / 2\ \-1 + 9*s /*(27 + 3*s)
$$\frac{\left(3 s^{2} + s\right) \left(s^{2} - 3 s + 9\right)}{\left(3 s + 27\right) \left(9 s^{2} - 1\right)}$$
(s + 3*s^2)*(9 + s^2 - 3*s)/((-1 + 9*s^2)*(27 + 3*s))
Numerical answer
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(s + 3.0*s^2)*(9.0 + s^2 - 3.0*s)/((27.0 + 3.0*s)*(-1.0 + 9.0*s^2))
(s + 3.0*s^2)*(9.0 + s^2 - 3.0*s)/((27.0 + 3.0*s)*(-1.0 + 9.0*s^2))
Combinatorics
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/ 2 \ s*\9 + s - 3*s/ -------------------- 3*(-1 + 3*s)*(9 + s)
$$\frac{s \left(s^{2} - 3 s + 9\right)}{3 \left(s + 9\right) \left(3 s - 1\right)}$$
s*(9 + s^2 - 3*s)/(3*(-1 + 3*s)*(9 + s))
Common denominator
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35 s -315 + 1018*s
- -- + - + --------------------
27 9 2
-243 + 81*s + 702*s
$$\frac{s}{9} + \frac{1018 s - 315}{81 s^{2} + 702 s - 243} - \frac{35}{27}$$
-35/27 + s/9 + (-315 + 1018*s)/(-243 + 81*s^2 + 702*s)
Combining rational expressions
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s*(1 + 3*s)*(9 + s*(-3 + s))
----------------------------
/ 2\
3*\-1 + 9*s /*(9 + s)
$$\frac{s \left(3 s + 1\right) \left(s \left(s - 3\right) + 9\right)}{3 \left(s + 9\right) \left(9 s^{2} - 1\right)}$$
s*(1 + 3*s)*(9 + s*(-3 + s))/(3*(-1 + 9*s^2)*(9 + s))
Powers
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/ 2\ / 2 \ \s + 3*s /*\9 + s - 3*s/ ------------------------- / 2\ \-1 + 9*s /*(27 + 3*s)
$$\frac{\left(3 s^{2} + s\right) \left(s^{2} - 3 s + 9\right)}{\left(3 s + 27\right) \left(9 s^{2} - 1\right)}$$
(s + 3*s^2)*(9 + s^2 - 3*s)/((-1 + 9*s^2)*(27 + 3*s))