## Algebra and Trigonometry 10th Edition

Published by Cengage Learning

# Chapter 3 - 3.4 - Zeroes of Polynomial Functions - 3.4 Exercises - Page 283: 19

x = -1, -2, 3

#### Work Step by Step

The Rational Zero Test can be used to find the rational zeros of the function. The Rational Zero Test relies on finding possible zeros and testing to see if they are actually zeros. The possible zeros are all the factors of the constant term divided by all the factors of the leading coefficient. f(x) = $x^{3}$ - 7x - 6 Factors of the constant: $\pm$ 1, $\pm$2, $\pm$3, $\pm$6 Factors of the leading coefficient: $\pm$1 All possible combinations: x = $\frac{\pm 1}{\pm 1}$, $\frac{\pm 2}{\pm 1}$, $\frac{\pm 3}{\pm 1}$, $\frac{\pm 6}{\pm 1}$ x = 1, -1, 2, -2, 3, -3, 6, -6 All of the possible combinations have to be tested: x = 1: f(1) = $1^{3}$ - 7(1) - 6 = -12 Since -12 $\ne$ 0, x = 1 is not a zero x = -1: f(-1) = $(-1)^{3}$ - 7(-1) - 6 = 0 Since 0 = 0, x = -1 is a zero x = 2: f(2) = $(2)^{3}$ - 7(2) - 6 = -12 Since -12 $\ne$ 0, x = 2 is not a zero x = -2: f(-2) = $(-2)^{3}$ - 7(-2) - 6 = 0 Since 0 = 0, x = -2 is a zero x = 3: f(3) = $(3)^{3}$ - 7(3) - 6 = 0 Since 0 = 0, x = 3 is a zero x = -3: f(-3) = $(-3)^{3}$ - 7(-3) - 6 = -12 Since -12 $\ne$ 0, x = -3 is not a zero x = 6: f(6) = $(6)^{3}$ - 7(6) - 6 = 168 Since 168 $\ne$ 0, x = 6 is not a zero x = -6: f(-6) = $(-6)^{3}$ - 7(-6) - 6 = -180 Since -180 $\ne$ 0, x = -6 is not a zero

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