SOLUTION: Find an​ nth-degree polynomial function with real coefficients satisfying the given conditions. n=3 4 and 4 i are​ zeros; f(1)= -102

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Question 1161653: Find an​ nth-degree polynomial function with real coefficients satisfying the given conditions.
n=3
4 and 4 i are​ zeros;
f(1)= -102
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
given :
n=3->degree
zeros:
x%5B1%5D=4
x%5B2%5D=4i -> complex solutions always come in pairs, so you also have
x%5B3%5D=-4i

f%28x%29=a%28x-x%5B1%5D%29%28x-x%5B2%5D%29%28x-x%5B3%5D%29

f%28x%29=a%28x-4%29%28x-4i%29%28x-%28-4i%29%29

f%28x%29=a%28x-4%29%28x-4i%29%28x%2B4i%29->%28x-4i%29%28x%2B4i%29=x%5E2-%284i%29%5E2=x%5E2-16%28-1%29=x%5E2%2B16

f%28x%29=a%28x-4%29%28x%5E2%2B16%29

f%28x%29=a%28x%5E3-4x%5E2%2B16x-64%29

then, use given

f%281%29=+-102 to calculate a
+-102=a%281%5E3-4%2A1%5E2%2B16%2A1-64%29
+-102=-51a
+a=-102%2F-51
+a=2
so, your equation is:
f%28x%29=2%28x%5E3-4x%5E2%2B16x-64%29
f%28x%29=2x%5E3-8x%5E2%2B32x-128