Find integers c and d such that the equation x^3+cx+d=0 has 1+sqrt(3) as one of its roots.

Makenna Booker 2022-07-27 Answered
Find integers c and d such that the equation x 3 + c x + d = 0 has 1 + 3 as one of its roots.
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Answers (1)

Anaya Gregory
Answered 2022-07-28 Author has 14 answers
Given an equation x 3 + b x 2 + c x + d that factors to ( x r 1 ) ( x r 2 ) ( x r 3 ),
b = r 1 + r 2 + r 3
c = r 1 r 2 + r 2 r 3 + r 3 r 1
d = r 1 r 2 r 3
For our problem, we'll say that r 1 = 1 + 3
In order for d to be an integer, either r 2 or r 3 must be ( 1 3 ) because we have to eliminate theradical in the same way we would if we were rationalizing thedenominator of a fraction.
So far we have r 1 = 1 + 3 , r 2 = 1 3 , and we know that r 1 + r 2 + r 3 = 0. Using these facts together you should be able to figure out what r 3 is and then plug into the above formul as to find c and d.
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