Question

See below.

Answer 1

where

Answer 2

Let \[a _{3}, a _{4}, ..., a _{2005}, a _{2006}\] be real numbers with \[a_{2006} \neq 0\]
Prove that there are not more than 2005 real numbers x such that
\[1+x+x^2+a_3x^3+a_4x^4+...+a_{2005}x^{2005}+a_{2006}x^{2006} = 0\]

Answer 3

So basically you want a proof that for any set of "a's" there will be at least 1 imaginary root

Answer 4

No, it could have a double root.

Answer 5

Bah i wish i could post a pic, it would be easier to explain my reasoning. Basically i created a vandermonde matrix with that function and showed it wasnt invertible, which means that some two rows of the matrix must be the same, which means some two roots must be equal.

Answer 6

i dont know if thats sufficient, but its all i could really think of.

Answer 7

http://en.wikipedia.org/wiki/Vandermonde_matrix
heres what the vandermonde matrix is.
Basically the property of this matrix im using is that its invertible if and only if each row is different, or rather, each root is different. But there is no way the matrix created by this problem can be invertible, because the vector (1, 1, 1, a_3, a_4,...., a_2006) is in its Null space. It has a non-trivial null space.