SOLUTION: Form a​ fifth-degree polynomial function with real coefficients such that 2i,1-2i, and -5 are zeros and f(0)=200.
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Question 1126935: Form a fifth-degree polynomial function with real coefficients such that 2i,1-2i, and -5 are zeros and f(0)=200.
Found 2 solutions by MathLover1, solver91311:
Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!
given:
, if that zero given, the conjugate must also be a zero
, we also have
recall: complex zeros always come in pairs
where is a constant, we put this for the y-intercept condition
Using the fact that , we can solve for .
so, we have
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Complex zeros always appear in conjugate pairs, thus if 
is a zero, then 
is also a zero. Hence the 5 zeros of your desired polynomial are: 
and 
.
If 
is a zero of a polynomial function )
, then 
is a factor of the polynomial. So, in factored form, your polynomial is:
\ =\ (x\ +\ 2i)(x\ -\ 2i)(x\ -\ 1\ -\ 2i)(x\ -\ 1\ +\ 2i)(x\ +\ 5))
Multiplying it out is left as an exercise for the student.
John
My calculator said it, I believe it, that settles it
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