deg=9;​​p[x_,n_]=Sum[Subscript[a,i]*x^i,{i,0,n}];​​q[x_,n_]=Sum[Subscript[b,i]*x^i,{i,0,n}];​​ϕ[x_]=p[x,deg]/q[x,deg]​​s={1,-2x,x^2};​​upper=Table[​​PadLeft[PadRight[s,deg+1-i,0],deg+1,0],{i,0,​​deg-2}];​​pcoeff=Table[Subscript[a,i],{i,0,deg}]//Reverse;​​qcoeff=Table[Subscript[b,i],{i,0,deg}]//Reverse;​​M=Join[upper,{pcoeff},{qcoeff}];​​M//MatrixForm​​Collect[Det[M],x]​​Collect[ϕ'[x]//Together//Numerator,x]​​Collect[ϕ'[x]//Together//Numerator,x]==Det[M]//Expand
Out[]=
a
0
+x
a
1
+
2
x
a
2
+
3
x
a
3
+
4
x
a
4
+
5
x
a
5
+
6
x
a
6
+
7
x
a
7
+
8
x
a
8
+
9
x
a
9
b
0
+x
b
1
+
2
x
b
2
+
3
x
b
3
+
4
x
b
4
+
5
x
b
5
+
6
x
b
6
+
7
x
b
7
+
8
x
b
8
+
9
x
b
9
Out[]//MatrixForm=
1
-2x
2
x
0
0
0
0
0
0
0
0
1
-2x
2
x
0
0
0
0
0
0
0
0
1
-2x
2
x
0
0
0
0
0
0
0
0
1
-2x
2
x
0
0
0
0
0
0
0
0
1
-2x
2
x
0
0
0
0
0
0
0
0
1
-2x
2
x
0
0
0
0
0
0
0
0
1
-2x
2
x
0
0
0
0
0
0
0
0
1
-2x
2
x
a
9
a
8
a
7
a
6
a
5
a
4
a
3
a
2
a
1
a
0
b
9
b
8
b
7
b
6
b
5
b
4
b
3
b
2
b
1
b
0
Out[]=
a
1
b
0
-
a
0
b
1
+x(2
a
2
b
0
-2
a
0
b
2
)+
2
x
(3
a
3
b
0
+
a
2
b
1
-
a
1
b
2
-3
a
0
b
3
)+
3
x
(4
a
4
b
0
+2
a
3
b
1
-2
a
1
b
3
-4
a
0
b
4
)+
4
x
(5
a
5
b
0
+3
a
4
b
1
+
a
3
b
2
-
a
2
b
3
-3
a
1
b
4
-5
a
0
b
5
)+
5
x
(6
a
6
b
0
+4
a
5
b
1
+2
a
4
b
2
-2
a
2
b
4
-4
a
1
b
5
-6
a
0
b
6
)+
6
x
(7
a
7
b
0
+5
a
6
b
1
+3
a
5
b
2
+
a
4
b
3
-
a
3
b
4
-3
a
2
b
5
-5
a
1
b
6
-7
a
0
b
7
)+
7
x
(8
a
8
b
0
+6
a
7
b
1
+4
a
6
b
2
+2
a
5
b
3
-2
a
3
b
5
-4
a
2
b
6
-6
a
1
b
7
-8
a
0
b
8
)+
8
x
(9
a
9
b
0
+7
a
8
b
1
+5
a
7
b
2
+3
a
6
b
3
+
a
5
b
4
-
a
4
b
5
-3
a
3
b
6
-5
a
2
b
7
-7
a
1
b
8
-9
a
0
b
9
)+
9
x
(8
a
9
b
1
+6
a
8
b
2
+4
a
7
b
3
+2
a
6
b
4
-2
a
4
b
6
-4
a
3
b
7
-6
a
2
b
8
-8
a
1
b
9
)+
10
x
(7
a
9
b
2
+5
a
8
b
3
+3
a
7
b
4
+
a
6
b
5
-
a
5
b
6
-3
a
4
b
7
-5
a
3
b
8
-7
a
2
b
9
)+
11
x
(6
a
9
b
3
+4
a
8
b
4
+2
a
7
b
5
-2
a
5
b
7
-4
a
4
b
8
-6
a
3
b
9
)+
12
x
(5
a
9
b
4
+3
a
8
b
5
+
a
7
b
6
-
a
6
b
7
-3
a
5
b
8
-5
a
4
b
9
)+
13
x
(4
a
9
b
5
+2
a
8
b
6
-2
a
6
b
8
-4
a
5
b
9
)+
14
x
(3
a
9
b
6
+
a
8
b
7
-
a
7
b
8
-3
a
6
b
9
)+
15
x
(2
a
9
b
7
-2
a
7
b
9
)+
16
x
(
a
9
b
8
-
a
8
b
9
)
Out[]=
a
1
b
0
-
a
0
b
1
+x(2
a
2
b
0
-2
a
0
b
2
)+
2
x
(3
a
3
b
0
+
a
2
b
1
-
a
1
b
2
-3
a
0
b
3
)+
3
x
(4
a
4
b
0
+2
a
3
b
1
-2
a
1
b
3
-4
a
0
b
4
)+
4
x
(5
a
5
b
0
+3
a
4
b
1
+
a
3
b
2
-
a
2
b
3
-3
a
1
b
4
-5
a
0
b
5
)+
5
x
(6
a
6
b
0
+4
a
5
b
1
+2
a
4
b
2
-2
a
2
b
4
-4
a
1
b
5
-6
a
0
b
6
)+
6
x
(7
a
7
b
0
+5
a
6
b
1
+3
a
5
b
2
+
a
4
b
3
-
a
3
b
4
-3
a
2
b
5
-5
a
1
b
6
-7
a
0
b
7
)+
7
x
(8
a
8
b
0
+6
a
7
b
1
+4
a
6
b
2
+2
a
5
b
3
-2
a
3
b
5
-4
a
2
b
6
-6
a
1
b
7
-8
a
0
b
8
)+
8
x
(9
a
9
b
0
+7
a
8
b
1
+5
a
7
b
2
+3
a
6
b
3
+
a
5
b
4
-
a
4
b
5
-3
a
3
b
6
-5
a
2
b
7
-7
a
1
b
8
-9
a
0
b
9
)+
9
x
(8
a
9
b
1
+6
a
8
b
2
+4
a
7
b
3
+2
a
6
b
4
-2
a
4
b
6
-4
a
3
b
7
-6
a
2
b
8
-8
a
1
b
9
)+
10
x
(7
a
9
b
2
+5
a
8
b
3
+3
a
7
b
4
+
a
6
b
5
-
a
5
b
6
-3
a
4
b
7
-5
a
3
b
8
-7
a
2
b
9
)+
11
x
(6
a
9
b
3
+4
a
8
b
4
+2
a
7
b
5
-2
a
5
b
7
-4
a
4
b
8
-6
a
3
b
9
)+
12
x
(5
a
9
b
4
+3
a
8
b
5
+
a
7
b
6
-
a
6
b
7
-3
a
5
b
8
-5
a
4
b
9
)+
13
x
(4
a
9
b
5
+2
a
8
b
6
-2
a
6
b
8
-4
a
5
b
9
)+
14
x
(3
a
9
b
6
+
a
8
b
7
-
a
7
b
8
-3
a
6
b
9
)+
15
x
(2
a
9
b
7
-2
a
7
b
9
)+
16
x
(
a
9
b
8
-
a
8
b
9
)
Out[]=
True