Math CentralQuandaries & Queries


Question from Kristen, a student:

I cannot figure out how to do this problem! I've been stuck on it forever. Please help!

Find a polynomial function with the indicated zeros and satisfying the given conditions. Simplify your answer (no imaginary numbers or parentheses in the answer)
Zeros: 1+2i, 1-2i, 5 ; f(-2)=1


Since 5, 1 + 2i and 1 - 2i are all zeros, (x - 5), (x - 1 - 2i) and (x - 1 + 2i) are all factors. Thus the polynomial must be of the form

f(x) = k(x - 5)(x - 1 - 2i)(x - 1 + 2i)

Expand the right side. [I would find (x - 1 - 2i)(x - 1 + 2i) first and then multiply by (x - 5).] What is the value of k is f(-2) = 1?


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