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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).

 May 25, 2021
 #1
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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

 

 

cool cool cool

 May 25, 2021

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