SOLUTION: Find the nth-degree polynomial function with real coefficients satisfying the given conditions n=4;i is a zero;-3 is a zero of multiplicity 2; f(-1)=16

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Question 1020535: Find the nth-degree polynomial function with real coefficients satisfying the given conditions
n=4;i is a zero;-3 is a zero of multiplicity 2; f(-1)=16
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
If i is a zero of the polynomial, then so is -i.
-3 a zero of multiplicity 2 means that %28x--3%29%5E2+=+%28x%2B3%29%5E2 is a factor of the polynomial.
Thus the polynomial has the factored form
a%5B0%5D%28x-i%29%28x%2Bi%29%28x%2B3%29%5E2+=+a%5B0%5D%28x%5E2%2B1%29%28x%2B3%29%5E2
for some constant a%5B0%5D.
To find a%5B0%5D, use f(-1) = 16.
==> a%5B0%5D%281%2B1%29%28-1%2B3%29%5E2+=+a%5B0%5D%2A2%2A2%5E2+=+8a%5B0%5D+=+16
==> a%5B0%5D+=+2
Thus the desired 4th degree polynomial is 2%28x%5E2%2B1%29%28x%2B3%29%5E2