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

• math8
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.
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?

have you tried induction on the degree of the polynomial?

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.

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