Re: Roots of Real Polynomials

Mate wrote:
Maury Barbato wrote:
David T. Ashley wrote:

"Maury Barbato" <mauriziobarbato@xxxxxxxx> wrote in
message
news:31996966.1166217070393.JavaMail.jakarta@nitrogen.
mathforum.org...
Hello,
a strange question crossed my mind.
Let P(x) a real polynomial. Does there exist a real
number c such that P(x)+c has all its roots in R?
I think the answer is negative, but I couldn't find
any
counterexample.
Thank you very much for your help.
My Best Regards,
Maury

Let me make the obvious observation that for a
second-order polynomial,
sqrt(B**2 - 4*A*C) says that one can always choose C
to place the roots in
R.

For higher-order polynomials ... I'm drawing a blank.





The statement holds for a third-order polynomial too,
as it can be verified using Cardano's Formula.

My Best Regards,
Maury

No, it does not. x^3 + x + c has a single real root for any real c.

I am trying to picture this in my head, and I am wondering if it is the
case, that the polynomials for which this works, are such that the
derivative has all real roots?

Taking your example,

d/dx(x^3 + x) = 3x^2 + 1, which has not real root.

-Darren

.

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