Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 2 - Section 2.5 - Zeros of Polynomial Functions - Exercise Set - Page 379: 70


The polynomial function \[{{x}^{4}}+6{{x}^{2}}+2=0\] does not have any rational roots.

Work Step by Step

According to the Rational Zero Theorem, Possible rational zeros = (Factors of the constant term)/ (Factors of the leading coefficient) ... (2) The constant term in equation (1) is 2. All the factors of 2 are $\pm 1\,\text{ and }\pm 2.$ The leading coefficient of the given polynomial in equation (1) is $+1$. Factors of the constant term 4 are $\pm 1\,\text{ and }\pm 2$. Factors of the leading coefficient are 1 and $\pm 1.$ Using the formula for the possible rational zeroes, we get that the possible rational zeroes are $\pm 1,\pm 2$. But from equation (1), we get that the minimum value of the polynomial is 2 and the terms have even power so on plotting the curve it will not intersect the x axis and it will always lie in the +x- axis. Hence, the polynomial function ${{x}^{4}}+6{{x}^{2}}+2=0$ will not have any rational roots.
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