Ohio Assessments for Educators (OAE) Mathematics Practice Exam

How are the roots of a polynomial characterized according to the Fundamental Theorem of Algebra?

1. As having a maximum equal to the degrees of the polynomial

2. As being unique and distinct

3. As potentially complex and irrational

4. As being limited to two for quadratic polynomials

The correct answer is: As having a maximum equal to the degrees of the polynomial

The correct response highlights that the roots of a polynomial can be characterized by their maximum quantity, which corresponds to the degree of the polynomial itself. According to the Fundamental Theorem of Algebra, a polynomial of degree \( n \) will have exactly \( n \) roots, when considering both real and complex roots, and counting multiplicities. This means that if a polynomial is of degree 3, for instance, it will have three roots, which could be a mix of real and complex numbers, but the total count of roots, including any repeated roots, will always equal the degree of the polynomial. The idea that roots can be unique or distinct is not necessarily true; roots may be repeated. Thus, they do not have to be unique. Considering that polynomials can indeed have complex and rational roots, this makes the third choice partially correct but insufficient as a comprehensive characterization of all roots. Lastly, while quadratic polynomials do have two roots, defining all polynomial roots solely in terms of quadratics is restrictive and overlooks the broader implications for polynomials of higher degrees. Therefore, stating that the number of roots corresponds to the degree of the polynomial is the most accurate characterization in the context of the Fundamental Theorem of Algebra.