SOLUTION: Find an nth- degree polynomial function with real coefficients satisfying the given conditions. n=3 3 and 5i are zeros f(-1) = -312 F(x)=

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Question 1178029: Find an nth- degree polynomial function with real coefficients satisfying the given conditions.
n=3
3 and 5i are zeros
f(-1) = -312
F(x)=
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Real coefficients with roots 3 and 5i means -5i is another root. Since the polynomial is degree 3, those are the only roots.

The linear factor corresponding to the root 3 is (x-3); the quadratic factor corresponding to the roots 5i and -5i is (x^2+25).

So f(x) is of the form

f%28x%29+=+a%28x-3%29%28x%5E2%2B25%29

Determine the constant a knowing that f(-1)=-312:

f%28-1%29+=+a%28-4%29%2826%29+=+-104a+=+-312
a+-312%2F-104+=+3

The function is

f%28x%29+=+-3%28x-3%29%28x%5E2%2B25%29
f%28x%29+=+-3x%5E3%2B9x%5E2-75x%2B225