Mister Exam
Factor polynomial x^4+2*x^3-x^2+2*x+1
The solution
You have entered
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4 3 2 x + 2*x - x + 2*x + 1
$$\left(2 x + \left(- x^{2} + \left(x^{4} + 2 x^{3}\right)\right)\right) + 1$$
x^4 + 2*x^3 - x^2 + 2*x + 1
Factorization
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/ ___\ / ___\ / ___\ / ___\ | 3 \/ 5 | | 3 \/ 5 | | 1 I*\/ 3 | | 1 I*\/ 3 | |x + - + -----|*|x + - - -----|*|x + - - + -------|*|x + - - - -------| \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$\left(x + \left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right)\right) \left(x + \left(\frac{\sqrt{5}}{2} + \frac{3}{2}\right)\right) \left(x + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)\right) \left(x + \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)\right)$$
(((x + 3/2 + sqrt(5)/2)*(x + 3/2 - sqrt(5)/2))*(x - 1/2 + i*sqrt(3)/2))*(x - 1/2 - i*sqrt(3)/2)
General simplification
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4 2 3 1 + x - x + 2*x + 2*x
$$x^{4} + 2 x^{3} - x^{2} + 2 x + 1$$
1 + x^4 - x^2 + 2*x + 2*x^3
Rational denominator
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4 2 3 1 + x - x + 2*x + 2*x
$$x^{4} + 2 x^{3} - x^{2} + 2 x + 1$$
1 + x^4 - x^2 + 2*x + 2*x^3
Powers
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4 2 3 1 + x - x + 2*x + 2*x
$$x^{4} + 2 x^{3} - x^{2} + 2 x + 1$$
1 + x^4 - x^2 + 2*x + 2*x^3
Trigonometric part
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4 2 3 1 + x - x + 2*x + 2*x
$$x^{4} + 2 x^{3} - x^{2} + 2 x + 1$$
1 + x^4 - x^2 + 2*x + 2*x^3
Assemble expression
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4 2 3 1 + x - x + 2*x + 2*x
$$x^{4} + 2 x^{3} - x^{2} + 2 x + 1$$
1 + x^4 - x^2 + 2*x + 2*x^3
Combinatorics
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/ 2 \ / 2 \ \1 + x - x/*\1 + x + 3*x/
$$\left(x^{2} - x + 1\right) \left(x^{2} + 3 x + 1\right)$$
(1 + x^2 - x)*(1 + x^2 + 3*x)
Common denominator
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4 2 3 1 + x - x + 2*x + 2*x
$$x^{4} + 2 x^{3} - x^{2} + 2 x + 1$$
1 + x^4 - x^2 + 2*x + 2*x^3
Combining rational expressions
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1 + x*(2 + x*(-1 + x*(2 + x)))
$$x \left(x \left(x \left(x + 2\right) - 1\right) + 2\right) + 1$$
1 + x*(2 + x*(-1 + x*(2 + x)))