Mister Exam

# Factor polynomial 1-a+a^2-a^3+a^4-a^5+a^6-a^7+a^8-a^9

An expression to simplify:

### The solution

You have entered [src]
         2    3    4    5    6    7    8    9
1 - a + a  - a  + a  - a  + a  - a  + a  - a 
$$- a^{9} + \left(a^{8} + \left(- a^{7} + \left(a^{6} + \left(- a^{5} + \left(a^{4} + \left(- a^{3} + \left(a^{2} + \left(1 - a\right)\right)\right)\right)\right)\right)\right)\right)$$
1 - a + a^2 - a^3 + a^4 - a^5 + a^6 - a^7 + a^8 - a^9
General simplification [src]
     2    4    6    8        3    5    7    9
1 + a  + a  + a  + a  - a - a  - a  - a  - a 
$$- a^{9} + a^{8} - a^{7} + a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1$$
1 + a^2 + a^4 + a^6 + a^8 - a - a^3 - a^5 - a^7 - a^9
Factorization [src]
        /                       ___________\ /                       ___________\ /                         ___________\ /                         ___________\ /                       ___________\ /                       ___________\ /                         ___________\ /                         ___________\
|          ___         /       ___ | |          ___         /       ___ | |            ___         /       ___ | |            ___         /       ___ | |          ___         /       ___ | |          ___         /       ___ | |            ___         /       ___ | |            ___         /       ___ |
|    1   \/ 5         /  5   \/ 5  | |    1   \/ 5         /  5   \/ 5  | |      1   \/ 5         /  5   \/ 5  | |      1   \/ 5         /  5   \/ 5  | |    1   \/ 5         /  5   \/ 5  | |    1   \/ 5         /  5   \/ 5  | |      1   \/ 5         /  5   \/ 5  | |      1   \/ 5         /  5   \/ 5  |
(a - 1)*|a + - - ----- + I*  /   - + ----- |*|a + - - ----- - I*  /   - + ----- |*|a + - - - ----- + I*  /   - - ----- |*|a + - - - ----- - I*  /   - - ----- |*|a + - + ----- + I*  /   - - ----- |*|a + - + ----- - I*  /   - - ----- |*|a + - - + ----- + I*  /   - + ----- |*|a + - - + ----- - I*  /   - + ----- |
\    4     4       \/    8     8   / \    4     4       \/    8     8   / \      4     4       \/    8     8   / \      4     4       \/    8     8   / \    4     4       \/    8     8   / \    4     4       \/    8     8   / \      4     4       \/    8     8   / \      4     4       \/    8     8   /
$$\left(a - 1\right) \left(a + \left(- \frac{\sqrt{5}}{4} + \frac{1}{4} + i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right) \left(a + \left(- \frac{\sqrt{5}}{4} + \frac{1}{4} - i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right) \left(a + \left(- \frac{\sqrt{5}}{4} - \frac{1}{4} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right) \left(a + \left(- \frac{\sqrt{5}}{4} - \frac{1}{4} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right) \left(a + \left(\frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right) \left(a + \left(\frac{1}{4} + \frac{\sqrt{5}}{4} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right) \left(a + \left(- \frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right) \left(a + \left(- \frac{1}{4} + \frac{\sqrt{5}}{4} - i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right)$$
((((((((a - 1)*(a + 1/4 - sqrt(5)/4 + i*sqrt(5/8 + sqrt(5)/8)))*(a + 1/4 - sqrt(5)/4 - i*sqrt(5/8 + sqrt(5)/8)))*(a - 1/4 - sqrt(5)/4 + i*sqrt(5/8 - sqrt(5)/8)))*(a - 1/4 - sqrt(5)/4 - i*sqrt(5/8 - sqrt(5)/8)))*(a + 1/4 + sqrt(5)/4 + i*sqrt(5/8 - sqrt(5)/8)))*(a + 1/4 + sqrt(5)/4 - i*sqrt(5/8 - sqrt(5)/8)))*(a - 1/4 + sqrt(5)/4 + i*sqrt(5/8 + sqrt(5)/8)))*(a - 1/4 + sqrt(5)/4 - i*sqrt(5/8 + sqrt(5)/8))
Common denominator [src]
     2    4    6    8        3    5    7    9
1 + a  + a  + a  + a  - a - a  - a  - a  - a 
$$- a^{9} + a^{8} - a^{7} + a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1$$
1 + a^2 + a^4 + a^6 + a^8 - a - a^3 - a^5 - a^7 - a^9
1.0 + a^2 + a^4 + a^6 + a^8 - a - a^3 - a^5 - a^7 - a^9
1.0 + a^2 + a^4 + a^6 + a^8 - a - a^3 - a^5 - a^7 - a^9
Combining rational expressions [src]
     2    4    6    8        3    5    7    9
1 + a  + a  + a  + a  - a - a  - a  - a  - a 
$$- a^{9} + a^{8} - a^{7} + a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1$$
1 + a^2 + a^4 + a^6 + a^8 - a - a^3 - a^5 - a^7 - a^9
Rational denominator [src]
     2    4    6    8        3    5    7    9
1 + a  + a  + a  + a  - a - a  - a  - a  - a 
$$- a^{9} + a^{8} - a^{7} + a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1$$
1 + a^2 + a^4 + a^6 + a^8 - a - a^3 - a^5 - a^7 - a^9
Combinatorics [src]
          /         2    3    4\ /     2    4        3\
-(-1 + a)*\1 + a + a  + a  + a /*\1 + a  + a  - a - a /
$$- \left(a - 1\right) \left(a^{4} - a^{3} + a^{2} - a + 1\right) \left(a^{4} + a^{3} + a^{2} + a + 1\right)$$
-(-1 + a)*(1 + a + a^2 + a^3 + a^4)*(1 + a^2 + a^4 - a - a^3)
Trigonometric part [src]
     2    4    6    8        3    5    7    9
1 + a  + a  + a  + a  - a - a  - a  - a  - a 
$$- a^{9} + a^{8} - a^{7} + a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1$$
1 + a^2 + a^4 + a^6 + a^8 - a - a^3 - a^5 - a^7 - a^9
Powers [src]
     2    4    6    8        3    5    7    9
1 + a  + a  + a  + a  - a - a  - a  - a  - a 
$$- a^{9} + a^{8} - a^{7} + a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1$$
1 + a^2 + a^4 + a^6 + a^8 - a - a^3 - a^5 - a^7 - a^9
Assemble expression [src]
     2    4    6    8        3    5    7    9
1 + a  + a  + a  + a  - a - a  - a  - a  - a 
$$- a^{9} + a^{8} - a^{7} + a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1$$
1 + a^2 + a^4 + a^6 + a^8 - a - a^3 - a^5 - a^7 - a^9
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