SOLUTION: How to write a polynomial of degree 3, with integer coefficients that has zeros √3i and 0

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Question 1181456: How to write a polynomial of degree 3, with integer coefficients that has zeros √3i and 0
Found 2 solutions by MathLover1, mananth:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

if a polynomial of degree 3 has zeros sqrt%283%29%2Ai and 0+, it also has zero -sqrt%283%29%2Ai (complex zeros always come in pairs)
then
f%28x%29=%28x-sqrt%283%29%2Ai%29%28x-%28-sqrt%283%29%2Ai%29%29%28x-0%29....expand
f%28x%29=%28x%5E2-%28sqrt%283%29%2Ai%29%5E2%29%28x%29
f%28x%29=%28x%5E2-3i%5E2%29%28x%29
f%28x%29=%28x%5E2-3%28-1%29%29%28x%29
f%28x%29=%28x%5E2%2B3%29%28x%29
f%28x%29=x%5E3%2B3x


Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!

To write a polynomial of degree 3 with specific roots x1 , x2 ,x3 write it out as
P(x) = (x−x1)(x−x2)(x−x3)
a polynomial (with real coefficients) must have complex roots in conjugate pairs so sqrt(3)i and -sqrt(3)i must be the roots , and 0 .
polynomial is
P(x) = x(x−sqrt(3)i)(x+sqrt(3)i)
=> x(x^2-3i^2)
=> x^3-(-3x)
x^3 +3x