1 | initial version |

The answer is similar to that one

First, when you write `6.00000000000000`

you define a numerical (floating-point approximated) number:

```
sage: 6.00000000000000.parent()
Real Field with 53 bits of precision
```

While, when you write `e^(-2/5*I*pi)`

, you define a symbolic expression:

```
sage: (e^(-2/5*I*pi)).parent()
Symbolic Ring
```

So, if you want something exact, you have to replace `6.00000000000000`

by `6`

, which is an exact integer:

```
sage: 6.parent()
Integer Ring
```

Then, your function is not a genuine polynomial but a symbolic expression (you could also have `cos(log(x))`

in there for example):

```
sage: p = (x^2 - 6)*x^2 - 4*x*(e^(2/5*I*pi) + e^(-2/5*I*pi)) - e^(4/5*I*pi) - e^(-4/5*I*pi) - 1
sage: p.parent()
Symbolic Ring
```

So, let us transform it into a genuine polynomial defined on the field of (complex) algebraic numbers:

```
sage: P = p.polynomial(QQbar)
sage: P.parent()
Univariate Polynomial Ring in x over Algebraic Field
```

The `Univariate Polynomial Ring in x over Algebraic Field`

makes more sense than a mysterious `Symbolic Ring`

, isn't it ?

In this safer place, you can have a look at its roots:

```
sage: P.roots()
[(-2.175570504584946? + 0.?e-17*I, 1),
(-0.618033988749895? + 0.?e-17*I, 1),
(0.1755705045849463? + 0.?e-18*I, 1),
(2.618033988749895? + 0.?e-17*I, 1)]
```

2 | No.2 Revision |

The answer is somehow similar to that one. It is important to know where your objects are living and define them into the right place since mathematics is not about manipulating ill-defined expressions.

First, when you write `6.00000000000000`

you define a numerical ~~(floating-point ~~(floating-point, i.e. approximated) number:

```
sage: 6.00000000000000.parent()
Real Field with 53 bits of precision
```

While, when you write `e^(-2/5*I*pi)`

, you define a symbolic expression:

```
sage: (e^(-2/5*I*pi)).parent()
Symbolic Ring
```

So, if you want something exact, you have to replace `6.00000000000000`

by `6`

, which is an exact integer:

```
sage: 6.parent()
Integer Ring
```

Also, `e^(-2/5*I*pi)`

is not any complex number, but an algebraic one, so you should take the benefit of this.

Then, your function is not a genuine polynomial but a symbolic expression (you could also have `cos(log(x))`

in there for example):

```
sage: p = (x^2 - 6)*x^2 - 4*x*(e^(2/5*I*pi) + e^(-2/5*I*pi)) - e^(4/5*I*pi) - e^(-4/5*I*pi) - 1
sage: p.parent()
Symbolic Ring
```

So, let us transform it into a genuine polynomial defined on the field of (complex) algebraic numbers:

```
sage: P = p.polynomial(QQbar)
sage: P.parent()
Univariate Polynomial Ring in x over Algebraic Field
```

The `Univariate Polynomial Ring in x over Algebraic Field`

makes more sense than a mysterious `Symbolic Ring`

, isn't it ?

In this safer place, you can have a look at its roots:

```
sage: P.roots()
[(-2.175570504584946? + 0.?e-17*I, 1),
(-0.618033988749895? + 0.?e-17*I, 1),
(0.1755705045849463? + 0.?e-18*I, 1),
(2.618033988749895? + 0.?e-17*I, 1)]
```

3 | No.3 Revision |

The answer is somehow similar to that one. It is important to know where your objects are living and define them into the right ~~place since mathematics is ~~place. As for mathematics, Sage will not ~~about manipulating ~~be able to handle ill-defined ~~expressions.~~expressions consistently.

First, when you write `6.00000000000000`

you define a numerical (floating-point, i.e. approximated) number:

```
sage: 6.00000000000000.parent()
Real Field with 53 bits of precision
```

While, when you write `e^(-2/5*I*pi)`

, you define a symbolic expression:

```
sage: (e^(-2/5*I*pi)).parent()
Symbolic Ring
```

So, if you want something exact, you have to replace `6.00000000000000`

by `6`

, which is an exact integer:

```
sage: 6.parent()
Integer Ring
```

Also, `e^(-2/5*I*pi)`

is not any complex number, but an algebraic one, so you should take the benefit of this.

Then, your function is not a genuine polynomial but a symbolic expression (you could also have `cos(log(x))`

in there for example):

```
sage: p = (x^2 - 6)*x^2 - 4*x*(e^(2/5*I*pi) + e^(-2/5*I*pi)) - e^(4/5*I*pi) - e^(-4/5*I*pi) - 1
sage: p.parent()
Symbolic Ring
```

So, let us transform it into a genuine polynomial defined on the field of (complex) algebraic numbers:

```
sage: P = p.polynomial(QQbar)
sage: P.parent()
Univariate Polynomial Ring in x over Algebraic Field
```

The `Univariate Polynomial Ring in x over Algebraic Field`

makes more sense than a mysterious `Symbolic Ring`

, isn't it ?

In this safer place, you can have a look at its roots:

```
sage: P.roots()
[(-2.175570504584946? + 0.?e-17*I, 1),
(-0.618033988749895? + 0.?e-17*I, 1),
(0.1755705045849463? + 0.?e-18*I, 1),
(2.618033988749895? + 0.?e-17*I, 1)]
```

4 | No.4 Revision |

The answer is somehow similar to that one. It is important to know where your objects are living and define them into the right place. As for mathematics, Sage will not be able to handle ill-defined expressions consistently.

First, when you write `6.00000000000000`

you define a numerical (floating-point, i.e. approximated) number:

```
sage: 6.00000000000000.parent()
Real Field with 53 bits of precision
```

While, when you write `e^(-2/5*I*pi)`

, you define a symbolic expression:

```
sage: (e^(-2/5*I*pi)).parent()
Symbolic Ring
```

`6.00000000000000`

by `6`

, which is an exact integer:

```
sage: 6.parent()
Integer Ring
```

Also, `e^(-2/5*I*pi)`

is not any complex number, but an algebraic one, so you should take the benefit of this.

Then, your function is not a genuine polynomial but a symbolic expression (you could also have `cos(log(x))`

in ~~there ~~such expression for example):

```
sage: p = (x^2 - 6)*x^2 - 4*x*(e^(2/5*I*pi) + e^(-2/5*I*pi)) - e^(4/5*I*pi) - e^(-4/5*I*pi) - 1
sage: p.parent()
Symbolic Ring
```

```
sage: P = p.polynomial(QQbar)
sage: P.parent()
Univariate Polynomial Ring in x over Algebraic Field
```

`Univariate Polynomial Ring in x over Algebraic Field`

makes more sense than a mysterious `Symbolic Ring`

, isn't it ?

In this safer place, you can have a look at its roots:

```
sage: P.roots()
[(-2.175570504584946? + 0.?e-17*I, 1),
(-0.618033988749895? + 0.?e-17*I, 1),
(0.1755705045849463? + 0.?e-18*I, 1),
(2.618033988749895? + 0.?e-17*I, 1)]
```

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