Ohio Assessments for Educators (OAE) Mathematics Practice Exam

According to the Fundamental Theorem of Algebra, what can be said about the roots of a polynomial?

1. The number of roots is at least equal to the degree of the polynomial

2. The number of roots is at most equal to the degree of the polynomial

3. Every polynomial has an equal number of real and imaginary roots

4. A polynomial cannot have more roots than its degree

The correct answer is: The number of roots is at most equal to the degree of the polynomial

The Fundamental Theorem of Algebra establishes that every non-constant polynomial function of degree n has exactly n roots in the complex number system, counting multiplicities. This means that for a polynomial of degree n, it will have precisely n roots when all possible instances of roots (both real and complex) are considered. Thus, while a polynomial may have fewer than n real roots, it will never exceed n when it comes to the total count of roots, including complex roots. Therefore, stating that the number of roots is at most equal to the degree of the polynomial accurately reflects this principle. In further context, other options present misleading interpretations or inaccuracies regarding the nature of polynomial roots. For example, suggesting that every polynomial has an equal number of real and imaginary roots does not reflect reality, as roots can be purely real, purely imaginary, or complex. The notion of having "at least" a certain number of roots does not align with the theorem, as some roots might not be real, thus not counted separately in certain contexts. Ultimately, the theorem firmly maintains that a polynomial cannot have a total number of roots exceeding its degree.