SOLUTION: Find an nth degree polynomial function with real coefficients satisfying the given conditions. n=3;-5 and i are zeros; f(-3)=60

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Question 1057890: Find an nth degree polynomial function with real coefficients satisfying the given conditions.
n=3;-5 and i are zeros; f(-3)=60
Answer by solve_for_x(190) About Me  (Show Source):
You can put this solution on YOUR website!
Since i is a zero, -i is also a zero, because imaginary roots always occur in conjugate pairs.

The factors of the desired polynomial are then (x - 5), (x - i), and (x + i).

The function is then:

f(x) = a(x - 5)(x - i)(x + i)

f(x) = a(x - 5)(x^2 - i^2)

f(x) = a(x - 5)(x^2 + 1)

f(x) = a(x^3 - 5x^2 + x - 5)

Then:

f(-3) = a((-3)^3 - 5(-3)^2 + (-3) - 5) = 60

a(-27 - 45 - 3 - 5) = 60

a(-80) = 60

a = -60/80

a = -3/4

The function is then:

f%28x%29+=+%28-3%2F4%29x%5E3+%2B+%2815%2F4%29x%5E2+-+%283%2F4%29x+%2B+15%2F4%29