- #1
math8
- 160
- 0
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?
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.
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.
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.
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.
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.