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## Operations on univariate polynomials

AXIOM SESSIONscale=2.5]

```N := NonNegativeInteger

(1)  NonNegativeInteger
Type: Domain
Time: 0 sec

Z := Integer

(2)  Integer
Type: Domain
Time: 0 sec

UZ := UnivariatePolynomial(x,Z)

(3)  UnivariatePolynomial(x,Integer)
Type: Domain
Time: 0 sec

u: UZ  := (3*x-1)**2 * (2*x + 8)

3      2
(4)  18x  + 60x  - 46x + 8
Type: UnivariatePolynomial(x,Integer)
Time: 0.01 (OT) = 0.01 sec

v: UZ := (1 - 6*x + 9*x**2)**2

4       3      2
(5)  81x  - 108x  + 54x  - 12x + 1
Type: UnivariatePolynomial(x,Integer)
Time: 0.01 (OT) = 0.01 sec

u**2 + u*v

7        6        5         4        3        2
(6)  1458x  + 3240x  - 7074x  + 10584x  - 9282x  + 4120x  - 878x + 72
Type: UnivariatePolynomial(x,Integer)
Time: 0.01 (OT) = 0.01 sec

(7)  18
Type: PositiveInteger
Time: 0 sec

degree u

(8)  3
Type: PositiveInteger
Time: 0 sec

reductum u

2
(9)  60x  - 46x + 8
Type: UnivariatePolynomial(x,Integer)
Time: 0 sec

gcd(u,v)

2
(10)  9x  - 6x + 1
Type: UnivariatePolynomial(x,Integer)
Time: 0 sec

factor(u)

2
(11)  2(x + 4)(3x - 1)
Type: Factored UnivariatePolynomial(x,Integer)
Time: 0 sec

factor(v)

4
(12)  (3x - 1)
Type: Factored UnivariatePolynomial(x,Integer)
Time: 0 sec

PZ := Polynomial Z

(13)  Polynomial Integer
Type: Domain
Time: 0 sec

pz: PZ := (4*x**3+2*y**2+1)*(12*x**5-x**3*y+12)

3 3       5       2        6    3        8      5      3
(14)  - 2x y  + (24x  + 24)y  + (- 4x  - x )y + 48x  + 12x  + 48x  + 12
Type: Polynomial Integer
Time: 0.01 (IN) = 0.01 sec

factor(pz)

3       5         2     3
(15)  - (x y - 12x  - 12)(2y  + 4x  + 1)
Type: Factored Polynomial Integer
Time: 0.04 (EV) = 0.04 sec

Q := Fraction Integer

(16)  Fraction Integer
Type: Domain
Time: 0 sec

PQ := Polynomial Q

(17)  Polynomial Fraction Integer
Type: Domain
Time: 0 sec

pq: PQ := (4*x**3+(2/3)*x**2+1)*(12*x**5-(1/2)*x**3+12)

8     7     6   35  5   95  3     2
(18)  48x  + 8x  - 2x  + -- x  + -- x  + 8x  + 12
3       2
Type: Polynomial Fraction Integer
Time: 0.03 (IN) + 0.01 (OT) = 0.04 sec

factor(pq)

3   1  2   1   5    1  3
(19)  48(x  + - x  + -)(x  - -- x  + 1)
6      4       24
Type: Factored Polynomial Fraction Integer
Time: 0.03 (EV) + 0.01 (OT) = 0.04 sec
```

AXIOM SESSIONscale=2.5]

```P := UnivariatePolynomial(x,Fraction Integer)

(1)  UnivariatePolynomial(x,Fraction Integer)
Type: Domain
Time: 0 sec

euclideanGcd(a,b) ==
l := [a];
a:=unitCanonical a
b:=unitCanonical b
while not zero? b repeat
l := cons(b,l)
(a,b):= (b,a rem b)
b:=unitCanonical b
reverse l

Type: Void
Time: 0 sec

p1: P  := x^8 + x^6  -3*x^4 -3*x^3 +8*x^2 +2*x -5

8    6     4     3     2
(3)  x  + x  - 3x  - 3x  + 8x  + 2x - 5
Type: UnivariatePolynomial(x,Fraction Integer)
Time: 0.01 (OT) = 0.01 sec
q1: P  := 3*x^6 + 5*x^4 - 4*x^2 -9*x +21

6     4     2
(4)  3x  + 5x  - 4x  - 9x + 21
Type: UnivariatePolynomial(x,Fraction Integer)
Time: 0.01 (IN) + 0.01 (OT) = 0.02 sec

euclideanGcd(p1,q1)

(5)
8    6     4     3     2            6   5  4   4  2
[x  + x  - 3x  - 3x  + 8x  + 2x - 5, x  + - x  - - x  - 3x + 7,
3      3
4   1  2   3   2   25     49      6150
x  - - x  + -, x  + -- x - --, x - ----, 1]
5      5       13     13      4663
Type: List UnivariatePolynomial(x,Fraction Integer)
Time: 0.01 (IN) + 0.01 (OT) = 0.02 sec

p2: P := x^30 + x^19 + 1

30    19
(6)  x   + x   + 1
Type: UnivariatePolynomial(x,Fraction Integer)
Time: 0 sec
q2: P := x^29 - x^17 + 3

29    17
(7)  x   - x   + 3
Type: UnivariatePolynomial(x,Fraction Integer)
Time: 0 sec

euclideanGcd(p2,q2)

(8)
30    19       29    17       19    18
[x   + x   + 1, x   - x   + 3, x   + x   - 3x + 1,

18    17     11     10     9     8     7     6     5     4     3     2
x   + x   - 3x   + 4x   - 4x  + 4x  - 4x  + 4x  - 4x  + 4x  - 4x  + 4x
+
- 4x - 2
,

12   4  11   4  10   4  9   4  8   4  7   4  6   4  5   4  4   4  3
x   - - x   + - x   - - x  + - x  - - x  + - x  - - x  + - x  - - x
3       3       3      3      3      3      3      3      3
+
4  2   1     1
- x  - - x + -
3      3     3
,

11   8732  10   8732  9   8732  8   10919  7   5816  6   10433  5
x   - ---- x   + ---- x  - ---- x  + ----- x  - ---- x  + ----- x
6545       6545      6545       6545      6545       6545
+
676  4   9164  3   8588  2   1756      878
- --- x  + ---- x  - ---- x  + ---- x + ----
595      6545      6545      1309     1309
,

10    9   19619  8   6553  7   15277  6       5       4      3      2
x   - x  - ----- x  - ---- x  - ----- x  - 729x  - 242x  - 82x  - 26x
16         4         16
+
59065     19555
- ----- x + -----
16        16
,
9    8   1  7   1  6     2        1
x  + x  + - x  + - x  + 3x  - 2x + -,
3      3                 3

8   78620  7   45799  6   34992  5   11616  4    4080  3    864   2
x  + ----- x  + ----- x  + ----- x  + ----- x  + ----- x  + ----- x
58777      58777      58777      58777      58777      58777
+
177403     58697
------ x - -----
58777     58777
,

7   170782  6   2173617  5   1746351  4   5658021  3   8599611  2
x  + ------ x  + ------- x  - ------- x  + ------- x  - ------- x
923785      3695140      3695140      3695140      3695140
+
2278869     1232437
------- x - -------
739028     1847570
,

6   2515753  5   15269857  4   12227447  3   2492039  2   2531080
x  - ------- x  + -------- x  - -------- x  + ------- x  - ------- x
142893       428679        428679        142893       428679
+
4446728
-------
428679
,
5   160644757  4   130375607  3   27115175  2   25927747     49124846
x  - --------- x  + --------- x  - -------- x  + -------- x - --------,
73785531       73785531      24595177      73785531     73785531
4   4553589485  3   1129145127  2   163361143     2886544997
x  - ---------- x  + ---------- x  + --------- x + ----------,
7507014112      3753507056      234594191     7507014112
3   112557717874  2   270014099776      15748937137
x  - ------------ x  + ------------ x - ------------,
43571774195      130715322585     130715322585
2   619536582958      472414012299      19773994490288
x  - ------------ x + -------------, x + --------------, 1]
633789802471     2535159209884      34258486832515
Type: List UnivariatePolynomial(x,Fraction Integer)
Time: 0.04 (OT) = 0.04 sec
```

Next: Algebraic systems Up: An introduction to Computer Algebra Previous: Operations on matrices
Marc Moreno Maza
2003-06-06