Let f(x) be a polynomial of degree 4 with rational coefficients which has 1 + 2*sqrt(3), 3 - sqrt(3) as roots, and such that f(0) = -154. Find f(1).
The other two roots are 1 - 2sqrt 3 3 + sqrt 3
We have this
P(x) = a [ x - (1 + sqrt 2) ] [ x - ( 1 - sqrt 2) ] [ x - (3 - sqrt 3) ] [ x - ( 3 + sqrt 3) ]
Expanding this we get
P(x) = a ( x^4 - 8 x^3 + 17 x^2 - 6 x - 6 )
Since f(0) = -154......then -6a = -154 ⇒ a = 154 / 6 = 77/3
So f(1) =
(77/3) ( 1 - 8 + 17 - 6 - 6) =
(77/3) ( -2) =
-154 / 3