Polynomial with at most n-1 solutions.

[peripatein]

Hi,

Homework Statement

I am expected to show that the polynomial
a₁x^b₁ + a₂x^b₂ + ... + a_nx^b_n = 0
has at most n-1 solutions in (0,infinity), where a₁,a₂,...,a_n are real numbers different than zero, and b₁,b₂,...,b_n are real numbers so that b_j is different than b_k for j different than k.

Homework Equations

The Attempt at a Solution

I am trying to apply Rolle's theorem, but am not very successful at that. In general, between any two solutions the first derivative is equal to zero.
I first tried dividing by x^b₁ noting that zero is not a solution and in order to obtain a simpler polynomial, but it is doubtful this is how it ought to be solved and it didn't seem to get me anywhere.
Would anyone kindly provide some further insight/guidance?

 

[Dick]

Hi,

Homework Statement

I am expected to show that the polynomial
a₁x^b₁ + a₂x^b₂ + ... + a_nx^b_n = 0
has at most n-1 solutions in (0,infinity), where a₁,a₂,...,a_n are real numbers different than zero, and b₁,b₂,...,b_n are real numbers so that b_j is different than b_k for j different than k.

Homework Equations

The Attempt at a Solution

I am trying to apply Rolle's theorem, but am not very successful at that. In general, between any two solutions the first derivative is equal to zero.
I first tried dividing by x^b₁ noting that zero is not a solution and in order to obtain a simpler polynomial, but it is doubtful this is how it ought to be solved and it didn't seem to get me anywhere.
Would anyone kindly provide some further insight/guidance?

Try thinking about the cases n=1 and n=2 first. You've got some good ideas there of using Rolle's theorem and dividing by x^(b1). Try and apply them to set up a proof by induction.