Which of the following properly describes the relationship between the degree of a polynomial and the number of zeros it has?

Answer 1

The relationship between the degree of a polynomial and the number of zeros it has can be described as follows:

1. A polynomial of degree \( n \) has exactly \( n \) roots (or zeros) if we count multiplicities and include complex roots. This means that if a root appears multiple times, it is counted that many times.

2. If we do not consider multiplicities, a polynomial of degree \( n \) may have fewer than \( n \) distinct zeros. Additionally, it may have zeros that are not real numbers (i.e., complex roots), especially if \( n \) is odd.

3. For real polynomials, if the degree is odd, there is at least one real root. If the degree is even, the polynomial may or may not have real roots, depending on its specific coefficients and constant term.

In summary, the main point is that a polynomial of degree \( n \) has up to \( n \) zeros, counting multiplicities and including complex zeros.