[Axiom-mail] Roots of Polynomials

[Gernot Hueber]

Hi, 

I would like to calculate the roots of polynomials with unknown or partly
unknown coefficients. For some rare cases there are solutions known e.g.
Cardans formulas for cubic polynomials, and low order reciprocal polynomials
can be solved.
However I don't know how to achieve this with Axiom.
See some trials below.
However, I guess I should be possible to get more than shown below, but I am
sorry I have no idea how. 

Thanks 

Gernot 

1) Second order polynomial 

(5) -> p1 := x**2 + a*x + b
(5) ->
     2
(5)  x  + a x + b
                                                 Type: Polynomial Integer
                                                              Time: 0 sec
(6) -> zerosOf(p1, x)
(6) ->
      +---------+        +---------+
      |        2         |        2
     \|- 4b + a   - a - \|- 4b + a   - a
(6)  [----------------,------------------]
             2                 2
                                            Type: List Expression Integer 

2) Cubic polynomial 

(9) -> p2 := x**3 + a*x**2 + b*x + c
(9) ->
     3      2
(9)  x  + a x  + b x + c
                                                 Type: Polynomial Integer
                                                 Time: 0.01(O) = 0.02 sec
(10) -> zerosOf(p2, x)
(10) ->
(10)
      +--------------------------+
      |      2                  2
     \|- 3%x9  - 2a %x9 - 4b + a   - %x9 - a
[%x9, ---------------------------------------,
                        2
   +--------------------------+
   |      2                  2
- \|- 3%x9  - 2a %x9 - 4b + a   - %x9 - a
-----------------------------------------]
                    2
                                            Type: List Expression Integer
                                                 Time: 0.02(E) = 0.02 sec
(11) -> definingPolynomial %x9
(11) ->
                    2       3
(11)  c + %x9 b + %x9 a + %x9
                                                 Type: Expression Integer
                                                              Time: 0 sec 

3) Simple reciprocal polynomial 

(12) -> p3 := x**3 + a*x**2 + a*x + 1
(12) ->
      3      2
(12)  x  + a x  + a x + 1
                                                 Type: Polynomial Integer
                                                              Time: 0 sec
(13) -> zerosOf(p3, x)
(13) ->
(13)
       +----------------------------+
       |       2              2
      \|- 3%x12  - 2a %x12 + a  - 4a  - %x12 - a
[%x12, ------------------------------------------,
                           2
   +----------------------------+
   |       2              2
- \|- 3%x12  - 2a %x12 + a  - 4a  - %x12 - a
--------------------------------------------]
                      2
                                            Type: List Expression Integer
                                                 Time: 0.01(O) = 0.02 sec
(14) -> definingPolynomial %x12
(14) ->
          2                3
(14)  (%x12  + %x12)a + %x12  + 1
                                                 Type: Expression Integer
                                                              Time: 0 sec
(15) -> rootsOf(p3, x)
(15) ->
(15)  [%x15,%x16,- %x16 - %x15 - a]
                                            Type: List Expression Integer
                                                 Time: 0.01(O) = 0.02 sec