Simplify a 12+ order symbolic determinant?

Questions
1

How can I simplify a 12+ order symbolic determination to its simplest form in mma?
The second example(In[2]) below should be a polynomial of x with maximal order of 12. But, mma gives a polynomial of x^192.

In[1]:    
b1 = RandomReal[{0, 1}, {11, 11}];    
c1 = Table[b1[[i, j]] + x, {i, 1, 11}, {j, 1, 11}];    
Print[c1 // MatrixForm];    
d = Simplify[Expand[Det[c1]]]   

Out[1]:    
0.033469 + 0.076227 x - 2.22045*10^-16 x^2 + 5.99675*10^-17 x^3 -  4.19082*10^-31 x^4 + 1.12616*10^-31 x^5 - 5.47382*10^-47 x^6 + 1.64215*10^-47 x^7

1.png
1.png1001×172 8.76 KB

In[2]:    
b1 = RandomReal[{0, 1}, {12, 12}];    
c1 = Table[b1[[i, j]] + x, {i, 1, 12}, {j, 1, 12}];    
d = Simplify[Expand[Det[c1]]]    

Out[2]:    
    `......` -8.17867*10^9 x^184 - 6.51297*10^8 x^185 - 4.49969*10^7 x^186 - 
    2.64212*10^6 x^187 - 128194. x^188 - 4934.19 x^189 - 
    141.247 x^190 - 2.67311 x^191 - 
    0.0250846 x^192)/(((0.214925 + x)^10) ((0.647147 + 
    1. x)^9) ((0.289144 + 1.56025 x + 1. x^2)^8) ......
2

Actually Det is giving a rational function result. The main issue is that Det will not be able to handle the mix of approximate values and symbolic expressions. One way to work around this is to interpolate the result.

3

I have no idea why we get a rational function instead of a polynomial. However, if we Rationalize the coefficients, the determinant becomes a polynomial of degree 1:

b1 = RandomReal[{0, 1}, {12, 12}];
c1 = b1 + x;
Det[Rationalize[c1, .000001]]
4

Symbolic Det uses one-step row reduction starting at dimension 12. With approximate coefficients it is not able to unravel what are supposed to be exact quotients. I know of some ways to address this, but none that I would trust for numeric reliability.