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Expression simplification/ Factorization polynomials/ x^4+x^2+x+3

Factor polynomial x^4+x^2+x+3

An expression to simplify:

The solution

You have entered [src] 
 4    2        
x  + x  + x + 3
$$\left(x + \left(x^{4} + x^{2}\right)\right) + 3$$ 
x^4 + x^2 + x + 3
General simplification [src] 
         2    4
3 + x + x  + x
$$x^{4} + x^{2} + x + 3$$ 
3 + x + x^2 + x^4
Factorization [src] 
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|                                                                                                 /     4        /    187   I*\/ 18627                                           2                                                        37              | |                                                                                                   /     4        /    187   I*\/ 18627                                           2                                                        37              | |                                                                                                   /     4        /    187   I*\/ 18627                                           2                                                        37              | |                  /     4        /    187   I*\/ 18627                                           2                                                        37                                                                                               |
|                                                                                                /    - - - 2*3 /   - --- + -----------  + ------------------------------------------------------------------------------ - ----------------------------- | |                                                                                                  /    - - - 2*3 /   - --- + -----------  - ------------------------------------------------------------------------------ - ----------------------------- | |                                                                                                  /    - - - 2*3 /   - --- + -----------  - ------------------------------------------------------------------------------ - ----------------------------- | |                 /    - - - 2*3 /   - --- + -----------  + ------------------------------------------------------------------------------ - -----------------------------                                                                                  |
|              ____________________________________________________________________             /       3     \/      432       144                  ____________________________________________________________________           _____________________ | |                ____________________________________________________________________             /       3     \/      432       144                  ____________________________________________________________________           _____________________ | |                ____________________________________________________________________             /       3     \/      432       144                  ____________________________________________________________________           _____________________ | |                /       3     \/      432       144                  ____________________________________________________________________           _____________________              ____________________________________________________________________|
|             /              _____________________                                             /                                                    /              _____________________                                           /             _______  | |               /              _____________________                                             /                                                    /              _____________________                                           /             _______  | |               /              _____________________                                             /                                                    /              _____________________                                           /             _______  | |               /                                                    /              _____________________                                           /             _______              /              _____________________                                 |
|            /              /             _______                                             /                                                    /              /             _______                                           /    187   I*\/ 18627   | |              /              /             _______                                             /                                                    /              /             _______                                           /    187   I*\/ 18627   | |              /              /             _______                                             /                                                    /              /             _______                                           /    187   I*\/ 18627   | |              /                                                    /              /             _______                                           /    187   I*\/ 18627              /              /             _______                                  |
|           /     2        /    187   I*\/ 18627                   37                        /                                                    /     2        /    187   I*\/ 18627                   37                 18*3 /   - --- + -----------  | |             /     2        /    187   I*\/ 18627                   37                        /                                                    /     2        /    187   I*\/ 18627                   37                 18*3 /   - --- + -----------  | |             /     2        /    187   I*\/ 18627                   37                        /                                                    /     2        /    187   I*\/ 18627                   37                 18*3 /   - --- + -----------  | |             /                                                    /     2        /    187   I*\/ 18627                   37                 18*3 /   - --- + -----------            /     2        /    187   I*\/ 18627                   37              |
|          /    - - + 2*3 /   - --- + -----------  + -----------------------------          /                                                    /    - - + 2*3 /   - --- + -----------  + -----------------------------       \/      432       144      | |            /    - - + 2*3 /   - --- + -----------  + -----------------------------          /                                                    /    - - + 2*3 /   - --- + -----------  + -----------------------------       \/      432       144      | |            /    - - + 2*3 /   - --- + -----------  + -----------------------------          /                                                    /    - - + 2*3 /   - --- + -----------  + -----------------------------       \/      432       144      | |            /                                                    /    - - + 2*3 /   - --- + -----------  + -----------------------------       \/      432       144               /    - - + 2*3 /   - --- + -----------  + ----------------------------- |
|         /       3     \/      432       144                _____________________         /                                                    /       3     \/      432       144                _____________________                                  | |           /       3     \/      432       144                _____________________         /                                                    /       3     \/      432       144                _____________________                                  | |           /       3     \/      432       144                _____________________         /                                                    /       3     \/      432       144                _____________________                                  | |           /                                                    /       3     \/      432       144                _____________________                                          /       3     \/      432       144                _____________________ |
|        /                                                  /             _______         /                                                    /                                                  /             _______                                   | |          /                                                  /             _______         /                                                    /                                                  /             _______                                   | |          /                                                  /             _______         /                                                    /                                                  /             _______                                   | |          /                                                    /                                                  /             _______                                          /                                                  /             _______  |
|       /                                                  /    187   I*\/ 18627         /                                                    /                                                  /    187   I*\/ 18627                                    | |         /                                                  /    187   I*\/ 18627         /                                                    /                                                  /    187   I*\/ 18627                                    | |         /                                                  /    187   I*\/ 18627         /                                                    /                                                  /    187   I*\/ 18627                                    | |         /                                                    /                                                  /    187   I*\/ 18627                                          /                                                  /    187   I*\/ 18627   |
|      /                                             18*3 /   - --- + -----------       /                                                    /                                             18*3 /   - --- + -----------                                   | |        /                                             18*3 /   - --- + -----------       /                                                    /                                             18*3 /   - --- + -----------                                   | |        /                                             18*3 /   - --- + -----------       /                                                    /                                             18*3 /   - --- + -----------                                   | |        /                                                    /                                             18*3 /   - --- + -----------                                        /                                             18*3 /   - --- + -----------  |
|    \/                                                 \/      432       144         \/                                                   \/                                                 \/      432       144                                       | |      \/                                                 \/      432       144         \/                                                   \/                                                 \/      432       144                                       | |      \/                                                 \/      432       144         \/                                                   \/                                                 \/      432       144                                       | |      \/                                                   \/                                                 \/      432       144                                          \/                                                 \/      432       144      |
|x + ------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|x + - ------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|x + - ------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|x + - -------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------|
\                                          2                                                                                                                           2                                                                                  / \                                            2                                                                                                                           2                                                                                  / \                                            2                                                                                                                           2                                                                                  / \                                                                                       2                                                                                                                           2                                       /
$$\left(x + \left(\frac{\sqrt{- \frac{4}{3} - 2 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}} - \frac{2}{\sqrt{- \frac{2}{3} + \frac{37}{18 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}} + 2 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}}} - \frac{37}{18 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}}}}{2} - \frac{\sqrt{- \frac{2}{3} + \frac{37}{18 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}} + 2 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}}}{2}\right)\right) \left(x + \left(\frac{\sqrt{- \frac{4}{3} - 2 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}} + \frac{2}{\sqrt{- \frac{2}{3} + \frac{37}{18 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}} + 2 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}}} - \frac{37}{18 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}}}}{2} + \frac{\sqrt{- \frac{2}{3} + \frac{37}{18 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}} + 2 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}}}{2}\right)\right) \left(x + \left(- \frac{\sqrt{- \frac{2}{3} + \frac{37}{18 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}} + 2 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}}}{2} - \frac{\sqrt{- \frac{4}{3} - 2 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}} - \frac{2}{\sqrt{- \frac{2}{3} + \frac{37}{18 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}} + 2 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}}} - \frac{37}{18 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}}}}{2}\right)\right) \left(x + \left(\frac{\sqrt{- \frac{2}{3} + \frac{37}{18 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}} + 2 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}}}{2} - \frac{\sqrt{- \frac{4}{3} - 2 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}} + \frac{2}{\sqrt{- \frac{2}{3} + \frac{37}{18 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}} + 2 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}}} - \frac{37}{18 \sqrt[3]{- \frac{187}{432} + \frac{\sqrt{18627} i}{144}}}}}{2}\right)\right)$$ 
(((x + sqrt(-2/3 + 2*(-187/432 + i*sqrt(18627)/144)^(1/3) + 37/(18*(-187/432 + i*sqrt(18627)/144)^(1/3)))/2 + sqrt(-4/3 - 2*(-187/432 + i*sqrt(18627)/144)^(1/3) + 2/sqrt(-2/3 + 2*(-187/432 + i*sqrt(18627)/144)^(1/3) + 37/(18*(-187/432 + i*sqrt(18627)/144)^(1/3))) - 37/(18*(-187/432 + i*sqrt(18627)/144)^(1/3)))/2)*(x - sqrt(-2/3 + 2*(-187/432 + i*sqrt(18627)/144)^(1/3) + 37/(18*(-187/432 + i*sqrt(18627)/144)^(1/3)))/2 + sqrt(-4/3 - 2*(-187/432 + i*sqrt(18627)/144)^(1/3) - 2/sqrt(-2/3 + 2*(-187/432 + i*sqrt(18627)/144)^(1/3) + 37/(18*(-187/432 + i*sqrt(18627)/144)^(1/3))) - 37/(18*(-187/432 + i*sqrt(18627)/144)^(1/3)))/2))*(x - sqrt(-2/3 + 2*(-187/432 + i*sqrt(18627)/144)^(1/3) + 37/(18*(-187/432 + i*sqrt(18627)/144)^(1/3)))/2 - sqrt(-4/3 - 2*(-187/432 + i*sqrt(18627)/144)^(1/3) - 2/sqrt(-2/3 + 2*(-187/432 + i*sqrt(18627)/144)^(1/3) + 37/(18*(-187/432 + i*sqrt(18627)/144)^(1/3))) - 37/(18*(-187/432 + i*sqrt(18627)/144)^(1/3)))/2))*(x - sqrt(-4/3 - 2*(-187/432 + i*sqrt(18627)/144)^(1/3) + 2/sqrt(-2/3 + 2*(-187/432 + i*sqrt(18627)/144)^(1/3) + 37/(18*(-187/432 + i*sqrt(18627)/144)^(1/3))) - 37/(18*(-187/432 + i*sqrt(18627)/144)^(1/3)))/2 + sqrt(-2/3 + 2*(-187/432 + i*sqrt(18627)/144)^(1/3) + 37/(18*(-187/432 + i*sqrt(18627)/144)^(1/3)))/2)
Trigonometric part [src] 
         2    4
3 + x + x  + x
$$x^{4} + x^{2} + x + 3$$ 
3 + x + x^2 + x^4
Numerical answer [src] 
3.0 + x + x^2 + x^4
3.0 + x + x^2 + x^4
Assemble expression [src] 
         2    4
3 + x + x  + x
$$x^{4} + x^{2} + x + 3$$ 
3 + x + x^2 + x^4
Common denominator [src] 
         2    4
3 + x + x  + x
$$x^{4} + x^{2} + x + 3$$ 
3 + x + x^2 + x^4
Combining rational expressions [src] 
      /      /     2\\
3 + x*\1 + x*\1 + x //
$$x \left(x \left(x^{2} + 1\right) + 1\right) + 3$$ 
3 + x*(1 + x*(1 + x^2))
Rational denominator [src] 
         2    4
3 + x + x  + x
$$x^{4} + x^{2} + x + 3$$ 
3 + x + x^2 + x^4
Combinatorics [src] 
         2    4
3 + x + x  + x
$$x^{4} + x^{2} + x + 3$$ 
3 + x + x^2 + x^4
Powers [src] 
         2    4
3 + x + x  + x
$$x^{4} + x^{2} + x + 3$$ 
3 + x + x^2 + x^4
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