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Expression simplification/ Factorization polynomials/ x^5+x+1

Factor polynomial x^5+x+1

An expression to simplify:

The solution

You have entered [src] 
 5        
x  + x + 1
$$\left(x^{5} + x\right) + 1$$ 
x^5 + x + 1
General simplification [src] 
         5
1 + x + x
$$x^{5} + x + 1$$ 
1 + x + x^5
Factorization [src] 
                                    /                                                        _______________                \ /                                                        _______________                \ /                                        _______________\
                                    |                                                       /          ____  /          ___\| |                                                       /          ____  /          ___\| |                                       /          ____ |
                                    |                                                      /  25   3*\/ 69   |  1   I*\/ 3 || |                                                      /  25   3*\/ 69   |  1   I*\/ 3 || |                                      /  25   3*\/ 69  |
/            ___\ /            ___\ |                                                   3 /   -- + -------- *|- - - -------|| |                                                   3 /   -- + -------- *|- - + -------|| |                                   3 /   -- + -------- |
|    1   I*\/ 3 | |    1   I*\/ 3 | |      1                     1                      \/    2       2      \  2      2   /| |      1                     1                      \/    2       2      \  2      2   /| |      1             1              \/    2       2     |
|x + - + -------|*|x + - - -------|*|x + - - + -------------------------------------- + ------------------------------------|*|x + - - + -------------------------------------- + ------------------------------------|*|x + - - + ---------------------- + --------------------|
\    2      2   / \    2      2   / |      3                          _______________                    3                  | |      3                          _______________                    3                  | |      3          _______________            3          |
                                    |            /          ___\     /          ____                                        | |            /          ___\     /          ____                                        | |                /          ____                        |
                                    |            |  1   I*\/ 3 |    /  25   3*\/ 69                                         | |            |  1   I*\/ 3 |    /  25   3*\/ 69                                         | |               /  25   3*\/ 69                         |
                                    |          3*|- - - -------|*3 /   -- + --------                                        | |          3*|- - + -------|*3 /   -- + --------                                        | |          3*3 /   -- + --------                        |
                                    \            \  2      2   / \/    2       2                                            / \            \  2      2   / \/    2       2                                            / \            \/    2       2                            /
$$\left(x + \left(\frac{1}{2} - \frac{\sqrt{3} i}{2}\right)\right) \left(x + \left(\frac{1}{2} + \frac{\sqrt{3} i}{2}\right)\right) \left(x + \left(- \frac{1}{3} + \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{25}{2}}}{3} + \frac{1}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{25}{2}}}\right)\right) \left(x + \left(- \frac{1}{3} + \frac{1}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{25}{2}}} + \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{25}{2}}}{3}\right)\right) \left(x + \left(- \frac{1}{3} + \frac{1}{3 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{25}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{25}{2}}}{3}\right)\right)$$ 
((((x + 1/2 + i*sqrt(3)/2)*(x + 1/2 - i*sqrt(3)/2))*(x - 1/3 + 1/(3*(-1/2 - i*sqrt(3)/2)*(25/2 + 3*sqrt(69)/2)^(1/3)) + (25/2 + 3*sqrt(69)/2)^(1/3)*(-1/2 - i*sqrt(3)/2)/3))*(x - 1/3 + 1/(3*(-1/2 + i*sqrt(3)/2)*(25/2 + 3*sqrt(69)/2)^(1/3)) + (25/2 + 3*sqrt(69)/2)^(1/3)*(-1/2 + i*sqrt(3)/2)/3))*(x - 1/3 + 1/(3*(25/2 + 3*sqrt(69)/2)^(1/3)) + (25/2 + 3*sqrt(69)/2)^(1/3)/3)
Rational denominator [src] 
         5
1 + x + x
$$x^{5} + x + 1$$ 
1 + x + x^5
Assemble expression [src] 
         5
1 + x + x
$$x^{5} + x + 1$$ 
1 + x + x^5
Numerical answer [src] 
1.0 + x + x^5
1.0 + x + x^5
Common denominator [src] 
         5
1 + x + x
$$x^{5} + x + 1$$ 
1 + x + x^5
Combining rational expressions [src] 
      /     4\
1 + x*\1 + x /
$$x \left(x^{4} + 1\right) + 1$$ 
1 + x*(1 + x^4)
Powers [src] 
         5
1 + x + x
$$x^{5} + x + 1$$ 
1 + x + x^5
Combinatorics [src] 
/         2\ /     3    2\
\1 + x + x /*\1 + x  - x /
$$\left(x^{2} + x + 1\right) \left(x^{3} - x^{2} + 1\right)$$ 
(1 + x + x^2)*(1 + x^3 - x^2)
Trigonometric part [src] 
         5
1 + x + x
$$x^{5} + x + 1$$ 
1 + x + x^5
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