SOLUTION: Find a 3rd degree polynomial with integral coefficients that has 2 +i and -3 as two of its roots.

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Question 801325: Find a 3rd degree polynomial with integral coefficients that has 2 +i and -3 as two of its roots.
Answer by josgarithmetic(39623) About Me  (Show Source):
You can put this solution on YOUR website!
Roots given ALSO include 2-i, the conjugate of 2+i.

Form the binomial factors of the polynomial based on the now known roots:
x+3, (x-(2+i)), (x-(2-i)).

Wanting only general form polynomial with integer coefficients, work with first the two complex factors with the imaginary components:
%28x-%282%2Bi%29%29%28x-%282-i%29%29
%28x-2-i%29%28x-2%2Bi%29
%28%28x-2%29-i%29%28%28x-2%29%2Bi%29
Recognize the factors giving the difference of two squares.
%28x-2%29%5E2-i%5E2
x%5E2-4x%2B4%2B1
x%5E2-4x%2B5

Still in factored form but showing only integer coefficients within each polynomial factor, polynomial is highlight%28%28x%2B3%29%28x%5E2-4x%2B5%29%29 which you can further multiply if you need.