Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
(s+1)/(3*s+6)*(5*s+1)/(s+3)-((1/10)*s+1)/((3/10)*s+1)
How do you (s+1)/(3*s+6)*(5*s+1)/(s+3)-((1/10)*s+1)/((3/10)*s+1) in partial fractions?
The solution
You have entered
[src]
s + 1 s
-------*(5*s + 1) -- + 1
3*s + 6 10
----------------- - -------
s + 3 3*s
--- + 1
10
$$\frac{\frac{s + 1}{3 s + 6} \left(5 s + 1\right)}{s + 3} - \frac{\frac{s}{10} + 1}{\frac{3 s}{10} + 1}$$
(((s + 1)/(3*s + 6))*(5*s + 1))/(s + 3) - (s/10 + 1)/(3*s/10 + 1)
General simplification
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3 2 -170 - 105*s + 12*s + 23*s ---------------------------- / 3 2 \ 3*\60 + 3*s + 25*s + 68*s/
$$\frac{12 s^{3} + 23 s^{2} - 105 s - 170}{3 \left(3 s^{3} + 25 s^{2} + 68 s + 60\right)}$$
(-170 - 105*s + 12*s^3 + 23*s^2)/(3*(60 + 3*s^3 + 25*s^2 + 68*s))
Fraction decomposition
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4/3 + 3/(2 + s) - 28/(3*(3 + s)) - 20/(3*(10 + 3*s))
$$\frac{4}{3} - \frac{20}{3 \left(3 s + 10\right)} - \frac{28}{3 \left(s + 3\right)} + \frac{3}{s + 2}$$
4 3 28 20 - + ----- - --------- - ------------ 3 2 + s 3*(3 + s) 3*(10 + 3*s)
Rational denominator
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(1 + s)*(1 + 5*s)*(100 + 30*s) + (-100 - 10*s)*(3 + s)*(6 + 3*s)
----------------------------------------------------------------
(3 + s)*(6 + 3*s)*(100 + 30*s)
$$\frac{\left(- 10 s - 100\right) \left(s + 3\right) \left(3 s + 6\right) + \left(s + 1\right) \left(5 s + 1\right) \left(30 s + 100\right)}{\left(s + 3\right) \left(3 s + 6\right) \left(30 s + 100\right)}$$
((1 + s)*(1 + 5*s)*(100 + 30*s) + (-100 - 10*s)*(3 + s)*(6 + 3*s))/((3 + s)*(6 + 3*s)*(100 + 30*s))
Combining rational expressions
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(1 + s)*(1 + 5*s)*(10 + 3*s) - 3*(2 + s)*(3 + s)*(10 + s)
---------------------------------------------------------
3*(2 + s)*(3 + s)*(10 + 3*s)
$$\frac{\left(s + 1\right) \left(3 s + 10\right) \left(5 s + 1\right) - 3 \left(s + 2\right) \left(s + 3\right) \left(s + 10\right)}{3 \left(s + 2\right) \left(s + 3\right) \left(3 s + 10\right)}$$
((1 + s)*(1 + 5*s)*(10 + 3*s) - 3*(2 + s)*(3 + s)*(10 + s))/(3*(2 + s)*(3 + s)*(10 + 3*s))
Powers
[src]
s
1 + --
10 (1 + s)*(1 + 5*s)
- ------- + -----------------
3*s (3 + s)*(6 + 3*s)
1 + ---
10
$$- \frac{\frac{s}{10} + 1}{\frac{3 s}{10} + 1} + \frac{\left(s + 1\right) \left(5 s + 1\right)}{\left(s + 3\right) \left(3 s + 6\right)}$$
s
-1 - --
10 (1 + s)*(1 + 5*s)
------- + -----------------
3*s (3 + s)*(6 + 3*s)
1 + ---
10
$$\frac{- \frac{s}{10} - 1}{\frac{3 s}{10} + 1} + \frac{\left(s + 1\right) \left(5 s + 1\right)}{\left(s + 3\right) \left(3 s + 6\right)}$$
(-1 - s/10)/(1 + 3*s/10) + (1 + s)*(1 + 5*s)/((3 + s)*(6 + 3*s))
Numerical answer
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-(1.0 + 0.1*s)/(1.0 + 0.3*s) + (1.0 + s)*(1.0 + 5.0*s)/((3.0 + s)*(6.0 + 3.0*s))
-(1.0 + 0.1*s)/(1.0 + 0.3*s) + (1.0 + s)*(1.0 + 5.0*s)/((3.0 + s)*(6.0 + 3.0*s))
Assemble expression
[src]
s
1 + --
10 (1 + s)*(1 + 5*s)
- ------- + -----------------
3*s (3 + s)*(6 + 3*s)
1 + ---
10
$$- \frac{\frac{s}{10} + 1}{\frac{3 s}{10} + 1} + \frac{\left(s + 1\right) \left(5 s + 1\right)}{\left(s + 3\right) \left(3 s + 6\right)}$$
-(1 + s/10)/(1 + 3*s/10) + (1 + s)*(1 + 5*s)/((3 + s)*(6 + 3*s))
Common denominator
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2
4 410 + 77*s + 377*s
- - --------------------------
3 3 2
180 + 9*s + 75*s + 204*s
$$- \frac{77 s^{2} + 377 s + 410}{9 s^{3} + 75 s^{2} + 204 s + 180} + \frac{4}{3}$$
4/3 - (410 + 77*s^2 + 377*s)/(180 + 9*s^3 + 75*s^2 + 204*s)
Trigonometric part
[src]
s
1 + --
10 (1 + s)*(1 + 5*s)
- ------- + -----------------
3*s (3 + s)*(6 + 3*s)
1 + ---
10
$$- \frac{\frac{s}{10} + 1}{\frac{3 s}{10} + 1} + \frac{\left(s + 1\right) \left(5 s + 1\right)}{\left(s + 3\right) \left(3 s + 6\right)}$$
-(1 + s/10)/(1 + 3*s/10) + (1 + s)*(1 + 5*s)/((3 + s)*(6 + 3*s))
Combinatorics
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3 2 -170 - 105*s + 12*s + 23*s ---------------------------- 3*(2 + s)*(3 + s)*(10 + 3*s)
$$\frac{12 s^{3} + 23 s^{2} - 105 s - 170}{3 \left(s + 2\right) \left(s + 3\right) \left(3 s + 10\right)}$$
(-170 - 105*s + 12*s^3 + 23*s^2)/(3*(2 + s)*(3 + s)*(10 + 3*s))