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Determine whether the following statement is true or false. If false, explain why. A polynomial function having degree 24 and only real coefficients may have no real zeros.

Determine whether the following statement is true or false. If false, explain why. A polynomial function having degree 24 and only real coefficients may have no real zeros.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Question:** Determine whether the following statement is true or false. If false, explain why. A polynomial function having degree 24 and only real coefficients may have no real zeros. --- **Analysis:** The statement concerns a polynomial function of degree 24 with real coefficients and questions whether it can have no real zeros. Polynomials with real coefficients and even degrees can indeed have no real zeros if their leading coefficient and their constant term are both positive (or both negative), causing the polynomial to remain above (or below) the x-axis. Examples include quadratics like \( f(x) = x^2 + 1 \) where no real x satisfies \( f(x) = 0 \). Therefore, the statement is **true**: a polynomial of even degree with real coefficients can have no real zeros.
Transcribed Image Text:**Question:** Determine whether the following statement is true or false. If false, explain why. A polynomial function having degree 24 and only real coefficients may have no real zeros. --- **Analysis:** The statement concerns a polynomial function of degree 24 with real coefficients and questions whether it can have no real zeros. Polynomials with real coefficients and even degrees can indeed have no real zeros if their leading coefficient and their constant term are both positive (or both negative), causing the polynomial to remain above (or below) the x-axis. Examples include quadratics like \( f(x) = x^2 + 1 \) where no real x satisfies \( f(x) = 0 \). Therefore, the statement is **true**: a polynomial of even degree with real coefficients can have no real zeros.
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