SOLUTION: 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

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
-----
f(x) = (x-5)(x-(-2+4i))(x-(-2-4i))
---
f(x) = (x-5)((x+2)-4i)((x+2)+4i)
---
f(x) = (x-5)((x+2)^2+16)
---
f(x) = (x-5)(x^2+4x+20)
===============================
Find a polynomial of degree 4 with Real Number coefficients and
with roots 3-2i, 2+i
---
f(x) = (x-(3-2i))(x-(3+2i))(x-(2+i))(x-(2-i))
---
f(x) = ((x-3)^2+4)((x-2)^2+1)
===================================
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:

John

My calculator said it, I believe it, that settles it

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