Determine whether each statement is true or false. If false, explain why. A polynomial function having degree 6 and only real coefficients may have no real zeros.

Verified step by step guidance

1. Identify the degree of the polynomial and the nature of its coefficients. In this case, the polynomial is of degree 6 and has only real coefficients.
2. Recall the Fundamental Theorem of Algebra, which states that a polynomial of degree n has exactly n roots in the complex number system.
3. Understand that roots of polynomials with real coefficients occur in conjugate pairs. This means if a complex number is a root, its conjugate is also a root.
4. Analyze the possible scenarios for the roots of a sixth-degree polynomial. Since the degree is even, the polynomial could potentially have all complex roots, all real roots, or a mix of both.
5. Conclude that it is possible for a sixth-degree polynomial with real coefficients to have no real zeros if all roots are complex numbers occurring in conjugate pairs.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial f(x) = 2x^6 - 3x^4 + 5, the degree is 6. Understanding polynomial functions is crucial for analyzing their properties, including the number and nature of their roots.

Real Coefficients

Real coefficients in a polynomial mean that all the coefficients of the terms are real numbers, as opposed to complex numbers. This characteristic influences the behavior of the polynomial's graph and its roots. For instance, a polynomial with real coefficients can have complex roots, but they must occur in conjugate pairs. This concept is essential for determining the possible zeros of the polynomial.

Roots and Zeros of Polynomials

The roots or zeros of a polynomial are the values of the variable that make the polynomial equal to zero. A polynomial of degree n can have up to n roots, but these can be real or complex. In the case of a polynomial with real coefficients, it is possible for a degree 6 polynomial to have no real zeros if all its roots are complex. This understanding is key to evaluating the truth of the statement in the question.