Show a polynomial of degree n has at most n distint roots

Thread starter math8
Start date May 18, 2008
Tags

Degree Polynomial Roots

In summary, a polynomial of degree n is a mathematical expression with variables, constants, and exponents, where the highest exponent determines the degree. A polynomial has distinct roots if each root is unique and does not repeat. The fundamental theorem of algebra states that a polynomial of degree n can have at most n complex roots. It is possible for a polynomial of degree n to have less than n distinct roots but not more than n distinct roots. The number of distinct roots is always equal to or less than the degree of the polynomial.

May 18, 2008

math8

If F is a field, how do we prove that a non-zero polynomial with coefficients in F and of degree n has at most n distinct roots in F?

Physics news on Phys.org

May 18, 2008

WlfordBrimley

have you tried induction on the degree of the polynomial?

May 18, 2008

mathwonk

Science Advisor

Homework Helper

e.g. can you do it for degree one polynomials?

1. What is a polynomial of degree n?

A polynomial of degree n is a mathematical expression that contains variables, constants, and exponents, and can be written in the form of ax^n + bx^(n-1) + ... + cx + d. The highest exponent in the polynomial determines its degree, with n being the highest possible degree.

2. What does it mean for a polynomial to have distinct roots?

A polynomial has distinct roots if each root is unique and does not repeat. In other words, if a polynomial of degree n has n distinct roots, it means that none of the roots are the same and there are no repeated roots in the polynomial.

3. How do you determine the number of roots a polynomial of degree n can have?

The fundamental theorem of algebra states that a polynomial of degree n can have at most n complex roots. This means that a polynomial of degree n can have n distinct roots at most.

4. Can a polynomial of degree n have less than n distinct roots?

Yes, a polynomial of degree n can have less than n distinct roots. However, it cannot have more than n distinct roots due to the fundamental theorem of algebra.

5. How is the number of distinct roots related to the degree of a polynomial?

The number of distinct roots of a polynomial is always equal to or less than the degree of the polynomial. This is because the degree of a polynomial represents the highest possible number of roots it can have, but it is not necessary for all polynomials to have the maximum number of roots.