Examples

Polynomials, rational functions:

coeff(X**3 + 3*X**2*Y + 3*X*Y**2 + Y**3,x);

  3    2
{Y ,3*Y ,3*Y,1}

gcd(X**2 + 4*X + 3,X**2 - 2*X - 3);

X + 1

resultant(X**2 + 4*X + 3,X**2 - 2*X - 3,x);

0

decompose(x**6+6x**4+x**3+9x**2+3x-5);

  2            3
{U  + U - 5,U=X  + 3*X}

factorize(x**6+6x**4+x**3+9x**2+3x);

    2      3
{X,X  + 3,X  + 3*X + 1}
 
roots(x**6+6x**4+x**3+9x**2+3x-5);

{X=0.543562, X=-0.775167, X=-0.27178+1.79488*I,
 X=-0.27178-1.79488*I, X=0.38758+1.8576*I,
 X=0.38758-1.8576*I}

interpol({0,7,26,63},z,{1,2,3,4});
 
 3
Z  - 1

pf(2/((x+1)^2*(x+2)),x); 

    2      - 2       2
{-------,-------,----------}
  X + 2   X + 1          2
                  (X + 1)

 
 m:=mat((1,x),(2,y));

     [1  X]
M := [    ]
     [2  Y]


 1/m;

[   - Y         X    ]
[---------  ---------]
[ 2*X - Y    2*X - Y ]
[                    ]
[    2         - 1   ]
[---------  ---------]
[ 2*X - Y    2*X - Y ]

 det m;

 - 2*X + Y

 
odesolve(df(y,x)=y+x**2+2,y,x);

    X                2
{Y=E *ARBCONST(1) - X  - 2*X - 4}

 solve({(a*x+y)/(z-1)-3,y+b+z,x-y},
       {x,y,z});

      - 3*(B + 1)
{{X=--------------,
        A + 4

      - 3*(B + 1)
  Y=--------------,
        A + 4

      - A*B - B + 3
  Z=----------------}}
         A + 4

 
 solve(a**(2*x)-3*a**x+2,x);

    2*ARBINT(2)*I*PI
{X=------------------,
         LOG(A)

    LOG(2) + 2*ARBINT(3)*I*PI
 X=---------------------------}
             LOG(A)

solve(
 { a*c1 - b*c1**2 - g*c1*c2 + e*c3,
   -g*c1*c2 + (e+t)*c3 -k*c2,
   g*c1*c2 + k*c2 - (e+t) * c3},
  {c3,c2,c1});

{{C1=ARBCOMPLEX(2),

   C1*(-C1*B*E-C1*B*T+A*E+A*T)
C2=---------------------------,
            C1*G*T-E*K

          2
   C1*(-C1 *B*G+C1*A*G-C1*B*K+A*K)
C3=-------------------------------}}
            C1*G*T-E*K

compact(s*(1-(sin x**2))
       +c*(1-(cos x)**2)
       +(sin x)**2+(cos x)**2,
   {cos x^2+sin x^2=1});

      2           2
SIN(X) *C + COS(X) *S + 1

 
 df(exp(x**2)/x,x,2);

     2
    X      4    2
 2*E  *(2*X  - X  + 1)
-----------------------
           3
          X

 int(x^3*exp(2x),x);

  2*X     3      2
 E   *(4*X  - 6*X  + 6*X - 3)
------------------------------
              8

 limit(x*sin(1/x),x,infinity);

1

on rounded; taylor(sin(x+1),x,0,4);

0.8414709848079 + 0.5403023058681*X
                                       3
- 0.420735492404*X - 0.09005038431136*X  
                    4
+ 0.03506129103366*X  + ...

sum(n,n);

 N*(N + 1)
-----------
     2

prod(n/(n+2),n);

      1
--------------
  2
 N  + 3*N + 2

 
 w:=(x+3*i)**2;

      2
W := X  + (6*I)*X - 9

 
precision 25; pi**2;

9.869 60440 10893 58618 83449 1

 
on modular; setmod 17; (x-1)**2;

 2
X  + 15*X + 1

factorize ws;

{X + 16,X + 16}