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Equations

[New] Problem :solve the polynomial equation .

```>> s:=solve(3*x^3-18*x^2+33*x-19):
>> simplify(rectform(s[i])) \$ i=1..3;

1/2     / PI \                      1/2     / PI \
2 3    cos | -- |                     3    cos | -- |
\ 18 /            / PI \            \ 18 /           / PI \
----------------- + 2, - sin | -- | - --------------- + 2 , sin | -- | -
3                    \ 18 /          3                  \ 18 /

1/2     / PI \
3    cos | -- |
\ 18 /
--------------- + 2
3
```

[New] Problem :solve the linear (dependent) system x+y+z=6,2x+y+2z=10,x+3y+z=10.

```>> linsolve({x+y+z=6,2*x+y+2*z=10,x+3*y+z=10},{x,y,z});

{y = 2, x = - z + 4 }
```

[New] Problem :solve the system of nonlinear equations .

```>> s:=solve({ x^2*y + 3*y*z - 4, -3*x^2*z + 2*y^2 + 1, 2*y*z^2 - z^2 - 1}):
>> subs(s, RootOf(18*z^8-48*z^7+21*z^6+5*z^4+3*z^2+1,z)=R);

{
{            /            2      3       4      5       6       7      2
{ x = RootOf \ 19 R - 48 R  + 2 R  + 48 R  + 3 R  - 48 R  + 18 R  + 6 x ,
{

2       4                            }
\        5 R    21 R        5      6             }
x /, y = - ---- - ----- + 24 R  - 9 R  - 1 , z = R }
2       2                              }
```

Here the solution is expressed with the help of the `RootOf` notation. There are 8 solutions, each one corresponding to z being one of the roots of the polynomial , y is a polynomial in z as shown above, and x being one root of a degree two polynomial dependent on z.

Andre Heck
Sun Apr 23 10:32:10 MDT 1995