Roots of derivative of polynomial.

In summary, the conversation is about proving that if all the roots of a polynomial P of order n>=2 are real, then all the roots of its derivative P' are also real. The person has tried using induction and Rolle's Mean Value Theorem but got stuck. They are asking for assistance and someone suggests considering two consecutive roots of P(x) and using a theorem to prove the statement. Eventually, the person manages to complete their proof.
  • #1
peripatein
880
0
Hi,

Homework Statement


I am asked to prove that given all roots of a polynomial P of order n>=2 are real, then all the roots of its derivative P' are necessarily real too.
I am permitted to assume that a polynomial of order n cannot have more than n real roots.

Homework Equations





The Attempt at a Solution


I have tried proving that using induction, but got stuck.
I'd appreciate some assistance.
 
Physics news on Phys.org
  • #2
Start by writing down P(x) in factored form.
 
  • #3
Think about two consecutive roots of P(x), say x1<x2. What can you say about P'(x) on the interval [x1,x2]? Try and find a theorem that applies.
 
  • #4
Based on Rolle's Mean Value Theorem, if P(x1)=P(x2)(=0, in this case) then P'(x3) = 0. I've tried using that but was unable to complete my proof by induction of the above mentioned statement.
Would you kindly assist?
 
  • #5
Won't P'(x) have n-1 real roots (number of xn+1-xn's)?
 
  • #6
I have managed, thanks :-)
 

FAQ: Roots of derivative of polynomial.

What is the definition of a polynomial?

A polynomial is a mathematical expression consisting of variables and coefficients, combined using only addition, subtraction, and multiplication. It can also include exponents, but the exponents must be non-negative integers.

What are the roots of a polynomial?

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, they are the solutions to the equation formed by setting the polynomial equal to zero.

How can I find the roots of a polynomial?

There are various methods for finding the roots of a polynomial, including factoring, graphing, or using the quadratic formula. The method used depends on the degree and complexity of the polynomial.

What is the derivative of a polynomial?

The derivative of a polynomial is a new polynomial that represents the rate of change of the original polynomial. It can be thought of as the slope of the tangent line to the polynomial curve at any given point.

How do I find the derivative of a polynomial?

The derivative of a polynomial can be found using the power rule, which states that the derivative of xn is nxn-1. The derivative of a polynomial is also equal to the sum of the derivatives of each term in the polynomial. In general, it is helpful to use the rules of differentiation to find the derivative of a polynomial.

Similar threads

Find interval and root of polynomial with absolute error less than 1/8
Replies
4
Views
197
Find this sum involving a polynomial root
Replies
7
Views
2K
Direct Proof that every zero of p(T) is an eigenvalue of T
Replies
24
Views
2K
Prove that roots of trig polynomials are denumerable
Replies
2
Views
2K
Showing a polynomial is not solvable by radicals
Replies
18
Views
4K
Cyclotomic Polynomial Questions
Replies
3
Views
1K
Proof that algebraic numbers are countable
Replies
6
Views
3K
Irreducibility and Roots in ##\Bbb{C}##
Replies
1
Views
1K
Finding the roots of a fourth degree polynomial
Replies
5
Views
2K
Factoring a quartic polynomial
Replies
12
Views
2K

Hot Threads

Recent Insights

Back
Top