Mister Exam
Factor polynomial x^4/2-x-x^2/2
The solution
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4 2 x x -- - x - -- 2 2
$$- \frac{x^{2}}{2} + \left(\frac{x^{4}}{2} - x\right)$$
x^4/2 - x - x^2/2
Factorization
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$$x \left(x + \left(- \frac{1}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{78}}{9} + 1}} - \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{78}}{9} + 1}\right)\right) \left(x + \left(- \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{78}}{9} + 1} - \frac{1}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{78}}{9} + 1}}\right)\right) \left(x + \left(- \sqrt[3]{\frac{\sqrt{78}}{9} + 1} - \frac{1}{3 \sqrt[3]{\frac{\sqrt{78}}{9} + 1}}\right)\right)$$
((x*(x - (1 + sqrt(78)/9)^(1/3)*(-1/2 - i*sqrt(3)/2) - 1/(3*(1 + sqrt(78)/9)^(1/3)*(-1/2 - i*sqrt(3)/2))))*(x - (1 + sqrt(78)/9)^(1/3)*(-1/2 + i*sqrt(3)/2) - 1/(3*(1 + sqrt(78)/9)^(1/3)*(-1/2 + i*sqrt(3)/2))))*(x - (1 + sqrt(78)/9)^(1/3) - 1/(3*(1 + sqrt(78)/9)^(1/3)))
Fraction decomposition
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x^4/2 - x - x^2/2
$$\frac{x^{4}}{2} - \frac{x^{2}}{2} - x$$
4 2 x x -- - x - -- 2 2
General simplification
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/ 3 \
x*\-2 + x - x/
---------------
2
$$\frac{x \left(x^{3} - x - 2\right)}{2}$$
x*(-2 + x^3 - x)/2
Trigonometric part
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4 2 x x -- - x - -- 2 2
$$\frac{x^{4}}{2} - \frac{x^{2}}{2} - x$$
x^4/2 - x - x^2/2
Common denominator
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4 2 x x -- - x - -- 2 2
$$\frac{x^{4}}{2} - \frac{x^{2}}{2} - x$$
x^4/2 - x - x^2/2
Combinatorics
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/ 3 \
x*\-2 + x - x/
---------------
2
$$\frac{x \left(x^{3} - x - 2\right)}{2}$$
x*(-2 + x^3 - x)/2
Assemble expression
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4 2 x x -- - x - -- 2 2
$$\frac{x^{4}}{2} - \frac{x^{2}}{2} - x$$
x^4/2 - x - x^2/2
Rational denominator
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4 2
x - x - 2*x
-------------
2
$$\frac{x^{4} - x^{2} - 2 x}{2}$$
(x^4 - x^2 - 2*x)/2
Combining rational expressions
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/ 3 \
x*\-2 + x - x/
---------------
2
$$\frac{x \left(x^{3} - x - 2\right)}{2}$$
x*(-2 + x^3 - x)/2