# Polynomial with at most n-1 solutions.

• peripatein
In summary, the conversation is discussing how to show that a polynomial has at most n-1 solutions in a specified interval, using Rolle's theorem and a proof by induction.
peripatein
Hi,

## Homework Statement

I am expected to show that the polynomial
a1xb1 + a2xb2 + ... + anxbn = 0
has at most n-1 solutions in (0,infinity), where a1,a2,...,an are real numbers different than zero, and b1,b2,...,bn are real numbers so that bj is different than bk for j different than k.

## 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 xb1 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?

peripatein said:
Hi,

## Homework Statement

I am expected to show that the polynomial
a1xb1 + a2xb2 + ... + anxbn = 0
has at most n-1 solutions in (0,infinity), where a1,a2,...,an are real numbers different than zero, and b1,b2,...,bn are real numbers so that bj is different than bk for j different than k.

## 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 xb1 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.

## What is a polynomial with at most n-1 solutions?

A polynomial with at most n-1 solutions is a mathematical expression consisting of variables and coefficients that can be solved for a finite number of values. The degree of the polynomial determines the maximum number of solutions it can have, with n-1 being the highest possible number of solutions.

## How do you determine the number of solutions for a polynomial?

The number of solutions for a polynomial can be determined by its degree. The degree of a polynomial is the highest power of its variable. For example, a polynomial with a degree of 3 can have at most 3 solutions. However, it is not guaranteed to have all 3 solutions.

## Can a polynomial have more than n-1 solutions?

Yes, a polynomial can have more than n-1 solutions. However, the degree of the polynomial determines the maximum number of solutions it can have. This means that a polynomial with a degree of 5 can have at most 5 solutions, but it can also have fewer solutions than that.

## Why is it important to know the number of solutions for a polynomial?

Knowing the number of solutions for a polynomial is important because it helps determine the behavior of the polynomial. A polynomial with a high number of solutions may have multiple turning points or intersections with other functions, while a polynomial with fewer solutions may have a simpler graph.

## How do you find the solutions for a polynomial with at most n-1 solutions?

The solutions for a polynomial with at most n-1 solutions can be found by setting the polynomial equal to zero and solving for the variable. This can be done using various methods such as factoring, the quadratic formula, or the rational roots theorem.

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