Let $f(x)$ be a polynomial of degree $4$ with rational coefficients. Find $f(1).$

The conjugates will also be  roots

So.....the  polynomial is

P(x)  =  a [x - ( 1+ 2sqrt (3))]  [ x - (1 - 2sqrt (3)) ]  [ x - (3-sqrt (2))] [ x - ( 3 + sqrt (2)) ]  =

(I used WolframAlpha to expand this  )

a (x^4 - 8 x^3 + 8 x^2 + 52 x - 77)

Since  f(0)  = -154

Then

a (-77)  = -154

a =  -154/-77  =   2

So....the polynomial  is

2 [ x^4 - 8 x^3 + 8 x^2 + 52 x - 77 ]  =

2x^4  - 16x^3 + 16x^2  + 104x - 154

And  we  just need to  add all the  coefficients plus the  constant term to get f(1)   =

2 - 16 + 16  + 104  - 154   = 

-48

You are correct!!!!!! Thanks for answering my question