Mister Exam

# Factor polynomial x^7+x^5+1

An expression to simplify:

### The solution

You have entered [src]
 7    5
x  + x  + 1
$$\left(x^{7} + x^{5}\right) + 1$$
x^7 + x^5 + 1
General simplification [src]
     5    7
1 + x  + x 
$$x^{7} + x^{5} + 1$$
1 + x^5 + x^7
Factorization [src]
/            ___\ /            ___\
|    1   I*\/ 3 | |    1   I*\/ 3 | /           / 5    4    3           \\ /           / 5    4    3           \\ /           / 5    4    3           \\ /           / 5    4    3           \\ /           / 5    4    3           \\
|x + - + -------|*|x + - - -------|*\x - CRootOf\x  - x  + x  - x + 1, 0//*\x - CRootOf\x  - x  + x  - x + 1, 1//*\x - CRootOf\x  - x  + x  - x + 1, 2//*\x - CRootOf\x  - x  + x  - x + 1, 3//*\x - CRootOf\x  - x  + x  - x + 1, 4//
\    2      2   / \    2      2   /                                                                                                                                                                                                   
$$\left(x + \left(\frac{1}{2} - \frac{\sqrt{3} i}{2}\right)\right) \left(x + \left(\frac{1}{2} + \frac{\sqrt{3} i}{2}\right)\right) \left(x - \operatorname{CRootOf} {\left(x^{5} - x^{4} + x^{3} - x + 1, 0\right)}\right) \left(x - \operatorname{CRootOf} {\left(x^{5} - x^{4} + x^{3} - x + 1, 1\right)}\right) \left(x - \operatorname{CRootOf} {\left(x^{5} - x^{4} + x^{3} - x + 1, 2\right)}\right) \left(x - \operatorname{CRootOf} {\left(x^{5} - x^{4} + x^{3} - x + 1, 3\right)}\right) \left(x - \operatorname{CRootOf} {\left(x^{5} - x^{4} + x^{3} - x + 1, 4\right)}\right)$$
((((((x + 1/2 + i*sqrt(3)/2)*(x + 1/2 - i*sqrt(3)/2))*(x - CRootOf(x^5 - x^4 + x^3 - x + 1, 0)))*(x - CRootOf(x^5 - x^4 + x^3 - x + 1, 1)))*(x - CRootOf(x^5 - x^4 + x^3 - x + 1, 2)))*(x - CRootOf(x^5 - x^4 + x^3 - x + 1, 3)))*(x - CRootOf(x^5 - x^4 + x^3 - x + 1, 4))
Powers [src]
     5    7
1 + x  + x 
$$x^{7} + x^{5} + 1$$
1 + x^5 + x^7
Common denominator [src]
     5    7
1 + x  + x 
$$x^{7} + x^{5} + 1$$
1 + x^5 + x^7
Trigonometric part [src]
     5    7
1 + x  + x 
$$x^{7} + x^{5} + 1$$
1 + x^5 + x^7
Combinatorics [src]
/         2\ /     3    5        4\
\1 + x + x /*\1 + x  + x  - x - x /
$$\left(x^{2} + x + 1\right) \left(x^{5} - x^{4} + x^{3} - x + 1\right)$$
(1 + x + x^2)*(1 + x^3 + x^5 - x - x^4)
Assemble expression [src]
     5    7
1 + x  + x 
$$x^{7} + x^{5} + 1$$
1 + x^5 + x^7
Combining rational expressions [src]
     5 /     2\
1 + x *\1 + x /
$$x^{5} \left(x^{2} + 1\right) + 1$$
1 + x^5*(1 + x^2)
1.0 + x^5 + x^7
1.0 + x^5 + x^7
     5    7
1 + x  + x 
$$x^{7} + x^{5} + 1$$
1 + x^5 + x^7