SOLUTION: 5+i and 1+4i Find a polynomial function f(x) of least degree having only real coefficients with zeros as given

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Question 1000695: 5+i and 1+4i
Find a polynomial function f(x) of least degree having only real coefficients with zeros as given
Found 2 solutions by ikleyn, MathLover1:
Answer by ikleyn(52794) (Show Source):
You can put this solution on YOUR website!
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If a polynomial with real coefficients has the zero 5+i, then it has 5-i as a zero too.

If a polynomial with real coefficients has the zero 1+4i, then it has 1-4i as a zero too.

Thus the polynomial of the minimal degree with real coefficients is

p(x) = (x-(5+i))*(x-(5-i))*(x-(1+4i))*(x-(1-4i)).

Now, (x-(5+i))*(x-(5-i)) = ,

(x-(1+4i))*(x-(1-4i)) = .

Hence,

p(x) = . .

You can open parentheses if you want/you need.

Answer by MathLover1(20850) (Show Source):
You can put this solution on YOUR website!

and
since given zeros are complex solutions, they always come in pairs
have and
then,

...multiplying each term by each, and knowing that , we have