Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
-1+x/(x^2-2*x+2)+(x^2+3*x+2)/(x^2-x+2)
How do you -1+x/(x^2-2*x+2)+(x^2+3*x+2)/(x^2-x+2) in partial fractions?
The solution
You have entered
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2
x x + 3*x + 2
-1 + ------------ + ------------
2 2
x - 2*x + 2 x - x + 2
$$\left(\frac{x}{\left(x^{2} - 2 x\right) + 2} - 1\right) + \frac{\left(x^{2} + 3 x\right) + 2}{\left(x^{2} - x\right) + 2}$$
-1 + x/(x^2 - 2*x + 2) + (x^2 + 3*x + 2)/(x^2 - x + 2)
General simplification
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/ 2\
x*\10 - 9*x + 5*x /
--------------------------
4 3 2
4 + x - 6*x - 3*x + 6*x
$$\frac{x \left(5 x^{2} - 9 x + 10\right)}{x^{4} - 3 x^{3} + 6 x^{2} - 6 x + 4}$$
x*(10 - 9*x + 5*x^2)/(4 + x^4 - 6*x - 3*x^3 + 6*x^2)
Fraction decomposition
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x/(2 + x^2 - 2*x) + 4*x/(2 + x^2 - x)
$$\frac{4 x}{x^{2} - x + 2} + \frac{x}{x^{2} - 2 x + 2}$$
x 4*x
------------ + ----------
2 2
2 + x - 2*x 2 + x - x
Rational denominator
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/ 2 \ / 2 \ / 2 \ / 2 \
\-2 - x + 3*x/*\2 + x - x/ + \2 + x - 2*x/*\2 + x + 3*x/
------------------------------------------------------------
/ 2 \ / 2 \
\2 + x - x/*\2 + x - 2*x/
$$\frac{\left(- x^{2} + 3 x - 2\right) \left(x^{2} - x + 2\right) + \left(x^{2} - 2 x + 2\right) \left(x^{2} + 3 x + 2\right)}{\left(x^{2} - 2 x + 2\right) \left(x^{2} - x + 2\right)}$$
((-2 - x^2 + 3*x)*(2 + x^2 - x) + (2 + x^2 - 2*x)*(2 + x^2 + 3*x))/((2 + x^2 - x)*(2 + x^2 - 2*x))
Common denominator
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2 3
- 9*x + 5*x + 10*x
--------------------------
4 3 2
4 + x - 6*x - 3*x + 6*x
$$\frac{5 x^{3} - 9 x^{2} + 10 x}{x^{4} - 3 x^{3} + 6 x^{2} - 6 x + 4}$$
(-9*x^2 + 5*x^3 + 10*x)/(4 + x^4 - 6*x - 3*x^3 + 6*x^2)
Assemble expression
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2
x 2 + x + 3*x
-1 + ------------ + ------------
2 2
2 + x - 2*x 2 + x - x
$$\frac{x}{x^{2} - 2 x + 2} - 1 + \frac{x^{2} + 3 x + 2}{x^{2} - x + 2}$$
-1 + x/(2 + x^2 - 2*x) + (2 + x^2 + 3*x)/(2 + x^2 - x)
Combinatorics
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/ 2\
x*\10 - 9*x + 5*x /
---------------------------
/ 2 \ / 2 \
\2 + x - x/*\2 + x - 2*x/
$$\frac{x \left(5 x^{2} - 9 x + 10\right)}{\left(x^{2} - 2 x + 2\right) \left(x^{2} - x + 2\right)}$$
x*(10 - 9*x + 5*x^2)/((2 + x^2 - x)*(2 + x^2 - 2*x))
Trigonometric part
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2
x 2 + x + 3*x
-1 + ------------ + ------------
2 2
2 + x - 2*x 2 + x - x
$$\frac{x}{x^{2} - 2 x + 2} - 1 + \frac{x^{2} + 3 x + 2}{x^{2} - x + 2}$$
-1 + x/(2 + x^2 - 2*x) + (2 + x^2 + 3*x)/(2 + x^2 - x)
Powers
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2
x 2 + x + 3*x
-1 + ------------ + ------------
2 2
2 + x - 2*x 2 + x - x
$$\frac{x}{x^{2} - 2 x + 2} - 1 + \frac{x^{2} + 3 x + 2}{x^{2} - x + 2}$$
-1 + x/(2 + x^2 - 2*x) + (2 + x^2 + 3*x)/(2 + x^2 - x)
Numerical answer
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-1.0 + x/(2.0 + x^2 - 2.0*x) + (2.0 + x^2 + 3.0*x)/(2.0 + x^2 - x)
-1.0 + x/(2.0 + x^2 - 2.0*x) + (2.0 + x^2 + 3.0*x)/(2.0 + x^2 - x)
Combining rational expressions
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(2 + x*(-1 + x))*(-2 + x - x*(-2 + x)) + (2 + x*(-2 + x))*(2 + x*(3 + x))
-------------------------------------------------------------------------
(2 + x*(-1 + x))*(2 + x*(-2 + x))
$$\frac{\left(x \left(x - 2\right) + 2\right) \left(x \left(x + 3\right) + 2\right) + \left(x \left(x - 1\right) + 2\right) \left(- x \left(x - 2\right) + x - 2\right)}{\left(x \left(x - 2\right) + 2\right) \left(x \left(x - 1\right) + 2\right)}$$
((2 + x*(-1 + x))*(-2 + x - x*(-2 + x)) + (2 + x*(-2 + x))*(2 + x*(3 + x)))/((2 + x*(-1 + x))*(2 + x*(-2 + x)))