Question 549764: I am having trouble can you help
Find a polynomial of degree 3 with roots 5, -2+4i
Find a polynomial of degree 4 with roots 3-2i, 2+i
thanks so much for your help
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Find a polynomial of degree 3 with Real Number coefficients and
with roots 5, -2+4i
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f(x) = (x-5)(x-(-2+4i))(x-(-2-4i))
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f(x) = (x-5)((x+2)-4i)((x+2)+4i)
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f(x) = (x-5)((x+2)^2+16)
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f(x) = (x-5)(x^2+4x+20)
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Find a polynomial of degree 4 with Real Number coefficients and
with roots 3-2i, 2+i
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f(x) = (x-(3-2i))(x-(3+2i))(x-(2+i))(x-(2-i))
---
f(x) = ((x-3)^2+4)((x-2)^2+1)
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Cheers,
Stan H.
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
You are only supposed to put one problem per post. So what I will do is tell you how to solve all problems of this type instead of solving these specific ones.
First: A polynomial function of degree  ALWAYS has roots.
Second: Irrational and complex roots ALWAYS come in conjugate pairs. That is, if  where  ,  , and  are real numbers,  , and is not a perfect square, then  is also a root. Also, if  is a root, then  is also a root.
Third: If  is a root, then  is a factor of the polynomial. So if the roots of an degree polynomial are the set  , then the completely factored polynomial is given by:
\ =\ \left(x\ -\ \alpha_1\right)\left(x\ -\ \alpha_2\right)\,\cdots\,\left(x\ -\ \alpha_n\right))
John
My calculator said it, I believe it, that settles it
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