SOLUTION: Find a polynomial with the following zeros. 4, 3+2i

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Question 844795: Find a polynomial with the following zeros. 4, 3+2i
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Complex roots with imaginary components occur in conjugate pairs.
The root, 3-2i is also a root.

Polynomial with the three roots given is %28x-4%29%28x-%283-2i%29%29%28x-%283%2B2i%29%29, and you just need to simplify to put into general form.

The multiplication and simplification:
You can more or less use distributive property on each of the complex factors with their imaginary components.
%28x-4%29%28x-3%2B2i%29%28x-3-2i%29
And apply an association on each of those complex factors.
%28x-4%29%28%28x-3%29%2B2i%29%28%28x-3%29-2i%29
Which neighboring factors are multiplied first does not matter. Take advantage of the difference of square result for the two complex factors with the imaginary components:
%28x-4%29%28%28x-3%29%5E2-%282i%29%5E2%29
%28x-4%29%28x%5E2-6x%2B9-4%28-1%29%29
%28x-4%29%28x%5E2-6x%2B9%2B4%29
highlight%28%28x-4%29%28x%5E2-6x%2B13%29%29-----You can take this one step further and multiply the binomial by the quadratic and put fully into general form.

x%5E2%28x-4%29-6x%28x-4%29%2B13%28x-4%29
x%5E3-4x%5E2-6x%5E2%2B24x%2B13x-52
highlight%28highlight%28x%5E3-10x%5E2%2B37x-52%29%29