### 2.25.18 Sylvester matrix of two polynomials : `sylvester`

`sylvester` takes two polynomials as arguments.
`sylvester` returns the Sylvester matrix S of these polynomials.
If A(x)=∑i=0i=n aixi and B(x)=∑i=0i=mbixi are 2 polynomials, their Sylvester matrix S is a squared matrix of size `m+n` where `m=degree(B(x))` and `n=degree(A(x))`. The `m` first lines are made with the A(x) coefficients, so that :

⎛ ⎜ ⎜ ⎜ ⎝ |
s11=an | s12=an−1 | ⋯ | s1(n+1)=a0 | 0 | ⋯ | 0
s21=0 | s22=an | ⋯ | s2(n+1)=a1 | s2(n+2)=a0 | ⋯ | 0
⋮ | ⋮ | ⋮ | ⋱ | ⋮ | ⋱ | ⋮
sm1=0 | sm2=0 | ⋯ | sm(n+1)=am−1 | sm(n+2)=am−2 | ⋯ | a0
| ⎞ ⎟ ⎟ ⎟ ⎠

and the `n` further lines are made with the B(x) coefficients, so that :

⎛ ⎜ ⎜ ⎝ |
s(m+1)1=bm | s(m+1)2=bm−1 | ⋯ | s(m+1)(m+1)=b0 | 0 | ⋯ | 0
⋮ | ⋮ | ⋮ | ⋱ | ⋮ | ⋱ | ⋮
s(m+n)1=0 | s(m+n)2=0 | ⋯ | s(m+n)(m+1)=bn−1 | bn−2 | ⋯ | b0
| ⎞ ⎟ ⎟ ⎠

Input :

`sylvester(x```^```3-p*x+q,3*x```^```2-p,x)`

Output :

`[[1,0,-p,q,0],[0,1,0,-p,q],[3,0,-p,0,0], [0,3,0,-p,0],[0,0,3,0,-p]]`

Input :

`det([[1,0,-p,q,0],[0,1,0,-p,q],[3,0,-p,0,0], [0,3,0,-p,0],[0,0,3,0,-p]])`

Output :

`-4*p```^```3--27*q```^```2`