Asked by Arielt
Suppose that a polynomial function of degree 4 with rational coefficients has i and (-3 + square root of 3)as zeros find the other zeros
Answers
Answered by
drwls
There must be four roots total and complex numbers have conjugates that are roots. One other root is therefore
-i .
If -3 + sqrt3, is a root, the another root is -3 - sqrt3, as a consequence of the +/-sqrt(b^2 - 4ac) in the quadratic equation.
The polynomial must be a multiple of
(x^2 +1)(x +3 -sqrt3)(x +3 +sqrt3) = 0
(x^2+1)[(x+3)^2 -3] = 0
(x^2+1)(x^2 +6x +6) = 0
x^4 +6x^3 +7x^2 +6 = 0
-i .
If -3 + sqrt3, is a root, the another root is -3 - sqrt3, as a consequence of the +/-sqrt(b^2 - 4ac) in the quadratic equation.
The polynomial must be a multiple of
(x^2 +1)(x +3 -sqrt3)(x +3 +sqrt3) = 0
(x^2+1)[(x+3)^2 -3] = 0
(x^2+1)(x^2 +6x +6) = 0
x^4 +6x^3 +7x^2 +6 = 0
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