# N-fold root of a polynomial

• December 1st 2010, 09:30 AM
doug
N-fold root of a polynomial
Let be f a real polynomial that is the sum of n monomials. Prove that there is no nonzero, at least n-fold root of the polynomial.

Any help would be appreciated!
• December 1st 2010, 03:06 PM
Drexel28
Quote:

Originally Posted by doug
Let be f a real polynomial that is the sum of n monomials. Prove that there is no nonzero, at least n-fold root of the polynomial.

Any help would be appreciated!

What does $N$-fold mean? I assume this is multivariable polynomials, right?
• December 1st 2010, 08:17 PM
topspin1617
I think it just means a root of multiplicity n.

Start by assuming the polynomial has a nonzero constant term (note that in proving this statement, we may ignore any common factors the monomials have; see if you can understand why).

Then try to prove that $f(x)$ has an n-fold root if and only if $f^{\prime}(x)$ has an (n-1)-fold root, and using induction.