Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
(2*x^5-10*x^4+20*x^3-32*x^2+16*x)/(x^2-2*x)^4
How do you (2*x^5-10*x^4+20*x^3-32*x^2+16*x)/(x^2-2*x)^4 in partial fractions?
The solution
You have entered
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5 4 3 2
2*x - 10*x + 20*x - 32*x + 16*x
-----------------------------------
4
/ 2 \
\x - 2*x/
$$\frac{16 x + \left(- 32 x^{2} + \left(20 x^{3} + \left(2 x^{5} - 10 x^{4}\right)\right)\right)}{\left(x^{2} - 2 x\right)^{4}}$$
(2*x^5 - 10*x^4 + 20*x^3 - 32*x^2 + 16*x)/(x^2 - 2*x)^4
Fraction decomposition
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x^(-3) - 2/(-2 + x)^4 + 2/(-2 + x)^3 - 1/(2*(-2 + x)^2) - 1/(4*x) + 1/(4*(-2 + x))
$$\frac{1}{4 \left(x - 2\right)} - \frac{1}{2 \left(x - 2\right)^{2}} + \frac{2}{\left(x - 2\right)^{3}} - \frac{2}{\left(x - 2\right)^{4}} - \frac{1}{4 x} + \frac{1}{x^{3}}$$
1 2 2 1 1 1 -- - --------- + --------- - ----------- - --- + ---------- 3 4 3 2 4*x 4*(-2 + x) x (-2 + x) (-2 + x) 2*(-2 + x)
General simplification
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/ 4 3 2\
2*\8 + x - 16*x - 5*x + 10*x /
--------------------------------
3 4
x *(-2 + x)
$$\frac{2 \left(x^{4} - 5 x^{3} + 10 x^{2} - 16 x + 8\right)}{x^{3} \left(x - 2\right)^{4}}$$
2*(8 + x^4 - 16*x - 5*x^3 + 10*x^2)/(x^3*(-2 + x)^4)
Numerical answer
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0.0625*(2.0*x^5 + 16.0*x + 20.0*x^3 - 32.0*x^2 - 10.0*x^4)/(-x + 0.5*x^2)^4
0.0625*(2.0*x^5 + 16.0*x + 20.0*x^3 - 32.0*x^2 - 10.0*x^4)/(-x + 0.5*x^2)^4
Common denominator
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3 4 2 16 - 32*x - 10*x + 2*x + 20*x --------------------------------- 7 4 6 3 5 x - 32*x - 8*x + 16*x + 24*x
$$\frac{2 x^{4} - 10 x^{3} + 20 x^{2} - 32 x + 16}{x^{7} - 8 x^{6} + 24 x^{5} - 32 x^{4} + 16 x^{3}}$$
(16 - 32*x - 10*x^3 + 2*x^4 + 20*x^2)/(x^7 - 32*x^4 - 8*x^6 + 16*x^3 + 24*x^5)
Assemble expression
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2 4 5 3
- 32*x - 10*x + 2*x + 16*x + 20*x
-------------------------------------
4
/ 2 \
\x - 2*x/
$$\frac{2 x^{5} - 10 x^{4} + 20 x^{3} - 32 x^{2} + 16 x}{\left(x^{2} - 2 x\right)^{4}}$$
(-32*x^2 - 10*x^4 + 2*x^5 + 16*x + 20*x^3)/(x^2 - 2*x)^4
Rational denominator
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2 4 5 3
- 32*x - 10*x + 2*x + 16*x + 20*x
-------------------------------------
4
/ 2 \
\x - 2*x/
$$\frac{2 x^{5} - 10 x^{4} + 20 x^{3} - 32 x^{2} + 16 x}{\left(x^{2} - 2 x\right)^{4}}$$
(-32*x^2 - 10*x^4 + 2*x^5 + 16*x + 20*x^3)/(x^2 - 2*x)^4
Combining rational expressions
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2*(8 + x*(-16 + x*(10 + x*(-5 + x))))
-------------------------------------
3 4
x *(-2 + x)
$$\frac{2 \left(x \left(x \left(x \left(x - 5\right) + 10\right) - 16\right) + 8\right)}{x^{3} \left(x - 2\right)^{4}}$$
2*(8 + x*(-16 + x*(10 + x*(-5 + x))))/(x^3*(-2 + x)^4)
Trigonometric part
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2 4 5 3
- 32*x - 10*x + 2*x + 16*x + 20*x
-------------------------------------
4
/ 2 \
\x - 2*x/
$$\frac{2 x^{5} - 10 x^{4} + 20 x^{3} - 32 x^{2} + 16 x}{\left(x^{2} - 2 x\right)^{4}}$$
(-32*x^2 - 10*x^4 + 2*x^5 + 16*x + 20*x^3)/(x^2 - 2*x)^4
Powers
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2 4 5 3
- 32*x - 10*x + 2*x + 16*x + 20*x
-------------------------------------
4
/ 2 \
\x - 2*x/
$$\frac{2 x^{5} - 10 x^{4} + 20 x^{3} - 32 x^{2} + 16 x}{\left(x^{2} - 2 x\right)^{4}}$$
(-32*x^2 - 10*x^4 + 2*x^5 + 16*x + 20*x^3)/(x^2 - 2*x)^4
Combinatorics
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/ 4 3 2\
2*\8 + x - 16*x - 5*x + 10*x /
--------------------------------
3 4
x *(-2 + x)
$$\frac{2 \left(x^{4} - 5 x^{3} + 10 x^{2} - 16 x + 8\right)}{x^{3} \left(x - 2\right)^{4}}$$
2*(8 + x^4 - 16*x - 5*x^3 + 10*x^2)/(x^3*(-2 + x)^4)