Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
a^3-a^2-a-2/(a^3+1)*a^3-2*a^2+2*a-1/(a^3+a^2+a)*a^2+a/(a^2-3*a+2)
How do you a^3-a^2-a-2/(a^3+1)*a^3-2*a^2+2*a-1/(a^3+a^2+a)*a^2+a/(a^2-3*a+2) in partial fractions?
The solution
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2
3 2 2 3 2 a a
a - a - a - ------*a - 2*a + 2*a - ----------- + ------------
3 3 2 2
a + 1 a + a + a a - 3*a + 2
$$\frac{a}{\left(a^{2} - 3 a\right) + 2} + \left(- \frac{a^{2}}{a + \left(a^{3} + a^{2}\right)} + \left(2 a + \left(- 2 a^{2} + \left(- a^{3} \frac{2}{a^{3} + 1} + \left(- a + \left(a^{3} - a^{2}\right)\right)\right)\right)\right)\right)$$
a^3 - a^2 - a - 2/(a^3 + 1)*a^3 - 2*a^2 + 2*a - a^2/(a^3 + a^2 + a) + a/(a^2 - 3*a + 2)
Fraction decomposition
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-2 + a + a^3 - 1/(-1 + a) - 3*a^2 + 2/(-2 + a) + 2/(3*(1 + a)) - a/(1 + a + a^2) - 2*(-2 + a)/(3*(1 + a^2 - a))
$$a^{3} - 3 a^{2} + a - \frac{a}{a^{2} + a + 1} - \frac{2 \left(a - 2\right)}{3 \left(a^{2} - a + 1\right)} - 2 + \frac{2}{3 \left(a + 1\right)} - \frac{1}{a - 1} + \frac{2}{a - 2}$$
3 1 2 2 2 a 2*(-2 + a)
-2 + a + a - ------ - 3*a + ------ + --------- - ---------- - --------------
-1 + a -2 + a 3*(1 + a) 2 / 2 \
1 + a + a 3*\1 + a - a/
General simplification
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/ 2 9 8 6 4 5 7\
a*\1 + a + a - 5*a - 4*a - 3*a + 4*a + 4*a + 7*a /
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7 6
2 + a - a - 2*a
$$\frac{a \left(a^{9} - 5 a^{8} + 7 a^{7} - 4 a^{6} + 4 a^{5} + 4 a^{4} + a^{2} - 3 a + 1\right)}{a^{7} - 2 a^{6} - a + 2}$$
a*(1 + a^2 + a^9 - 5*a^8 - 4*a^6 - 3*a + 4*a^4 + 4*a^5 + 7*a^7)/(2 + a^7 - a - 2*a^6)
Combining rational expressions
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// 3\ / 3 / 2 3 / 3\ / 3\\\\
a*\\1 + a /*(1 + a*(1 + a)) + (2 + a*(-3 + a))*\-1 - a + (1 + a*(1 + a))*\2 - 2*a + 2*a + \1 + a /*(-1 + a*(-1 + a)) - 2*a*\1 + a ////
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/ 3\
\1 + a /*(1 + a*(1 + a))*(2 + a*(-3 + a))
$$\frac{a \left(\left(a^{3} + 1\right) \left(a \left(a + 1\right) + 1\right) + \left(a \left(a - 3\right) + 2\right) \left(- a^{3} + \left(a \left(a + 1\right) + 1\right) \left(2 a^{3} - 2 a^{2} - 2 a \left(a^{3} + 1\right) + \left(a^{3} + 1\right) \left(a \left(a - 1\right) - 1\right) + 2\right) - 1\right)\right)}{\left(a^{3} + 1\right) \left(a \left(a - 3\right) + 2\right) \left(a \left(a + 1\right) + 1\right)}$$
a*((1 + a^3)*(1 + a*(1 + a)) + (2 + a*(-3 + a))*(-1 - a^3 + (1 + a*(1 + a))*(2 - 2*a^2 + 2*a^3 + (1 + a^3)*(-1 + a*(-1 + a)) - 2*a*(1 + a^3))))/((1 + a^3)*(1 + a*(1 + a))*(2 + a*(-3 + a)))
Assemble expression
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2 3
3 2 a a 2*a
a + a - 3*a + ------------ - ----------- - ------
2 2 3 3
2 + a - 3*a a + a + a 1 + a
$$a^{3} - \frac{2 a^{3}}{a^{3} + 1} - 3 a^{2} - \frac{a^{2}}{a^{3} + a^{2} + a} + a + \frac{a}{a^{2} - 3 a + 2}$$
a + a^3 - 3*a^2 + a/(2 + a^2 - 3*a) - a^2/(a + a^2 + a^3) - 2*a^3/(1 + a^3)
Trigonometric part
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2 3
3 2 a a 2*a
a + a - 3*a + ------------ - ----------- - ------
2 2 3 3
2 + a - 3*a a + a + a 1 + a
$$a^{3} - \frac{2 a^{3}}{a^{3} + 1} - 3 a^{2} - \frac{a^{2}}{a^{3} + a^{2} + a} + a + \frac{a}{a^{2} - 3 a + 2}$$
a + a^3 - 3*a^2 + a/(2 + a^2 - 3*a) - a^2/(a + a^2 + a^3) - 2*a^3/(1 + a^3)
Rational denominator
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// 2 3\ / 3 / 3\ / 3 2\ 2 / 3\ / 3\\ 2 / 3\\ / 2 \ / 3\ / 2 3\
\\a + a + a /*\- 2*a + \1 + a /*\a - a - a / - 2*a *\1 + a / + 2*a*\1 + a // - a *\1 + a //*\2 + a - 3*a/ + a*\1 + a /*\a + a + a /
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/ 3\ / 2 \ / 2 3\
\1 + a /*\2 + a - 3*a/*\a + a + a /
$$\frac{a \left(a^{3} + 1\right) \left(a^{3} + a^{2} + a\right) + \left(- a^{2} \left(a^{3} + 1\right) + \left(a^{3} + a^{2} + a\right) \left(- 2 a^{3} - 2 a^{2} \left(a^{3} + 1\right) + 2 a \left(a^{3} + 1\right) + \left(a^{3} + 1\right) \left(a^{3} - a^{2} - a\right)\right)\right) \left(a^{2} - 3 a + 2\right)}{\left(a^{3} + 1\right) \left(a^{2} - 3 a + 2\right) \left(a^{3} + a^{2} + a\right)}$$
(((a + a^2 + a^3)*(-2*a^3 + (1 + a^3)*(a^3 - a - a^2) - 2*a^2*(1 + a^3) + 2*a*(1 + a^3)) - a^2*(1 + a^3))*(2 + a^2 - 3*a) + a*(1 + a^3)*(a + a^2 + a^3))/((1 + a^3)*(2 + a^2 - 3*a)*(a + a^2 + a^3))
Combinatorics
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/ 2 9 8 6 4 5 7\
a*\1 + a + a - 5*a - 4*a - 3*a + 4*a + 4*a + 7*a /
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/ 2\ / 2 \
(1 + a)*(-1 + a)*(-2 + a)*\1 + a + a /*\1 + a - a/
$$\frac{a \left(a^{9} - 5 a^{8} + 7 a^{7} - 4 a^{6} + 4 a^{5} + 4 a^{4} + a^{2} - 3 a + 1\right)}{\left(a - 2\right) \left(a - 1\right) \left(a + 1\right) \left(a^{2} - a + 1\right) \left(a^{2} + a + 1\right)}$$
a*(1 + a^2 + a^9 - 5*a^8 - 4*a^6 - 3*a + 4*a^4 + 4*a^5 + 7*a^7)/((1 + a)*(-1 + a)*(-2 + a)*(1 + a + a^2)*(1 + a^2 - a))
Powers
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2 3
3 2 a a 2*a
a + a - 3*a + ------------ - ----------- - ------
2 2 3 3
2 + a - 3*a a + a + a 1 + a
$$a^{3} - \frac{2 a^{3}}{a^{3} + 1} - 3 a^{2} - \frac{a^{2}}{a^{3} + a^{2} + a} + a + \frac{a}{a^{2} - 3 a + 2}$$
a + a^3 - 3*a^2 + a/(2 + a^2 - 3*a) - a^2/(a + a^2 + a^3) - 2*a^3/(1 + a^3)
Common denominator
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4 3 2 5
3 2 4 + a - 4*a - 3*a + 4*a + 4*a
-2 + a + a - 3*a + ---------------------------------
7 6
2 + a - a - 2*a
$$a^{3} - 3 a^{2} + a - 2 + \frac{4 a^{5} + a^{4} - 4 a^{3} + 4 a^{2} - 3 a + 4}{a^{7} - 2 a^{6} - a + 2}$$
-2 + a + a^3 - 3*a^2 + (4 + a^4 - 4*a^3 - 3*a + 4*a^2 + 4*a^5)/(2 + a^7 - a - 2*a^6)
Numerical answer
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a^3 + 1.0*a - 3.0*a^2 + a/(2.0 + a^2 - 3.0*a) - a^2/(a + a^2 + a^3) - 2.0*a^3/(1.0 + a^3)
a^3 + 1.0*a - 3.0*a^2 + a/(2.0 + a^2 - 3.0*a) - a^2/(a + a^2 + a^3) - 2.0*a^3/(1.0 + a^3)