Use the Rational Zero Theorem to list all possible rational zeros for the given function.
-$$f ( x ) = - 2 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 8$$
A) $\pm \frac { 1 } { 2 } , \pm 1 , \pm 2 , \pm 4 , \pm 8$
B) $\pm \frac { 1 } { 4 } , \pm \frac { 1 } { 2 } , \pm 1 , \pm 2 , \pm 4 , \pm 8$
C) $\pm \frac { 1 } { 8 } , \pm \frac { 1 } { 4 } , \pm \frac { 1 } { 2 } , \pm 1 , \pm 2 , \pm 4 , \pm 8$
D) $\pm \frac { 1 } { 2 } , \pm 1 , \pm 2 , \pm 4$

Question

Quizplus · Accepted Answer

The Rational Zero Theorem states that any rational zero of a polynomial function $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ can be expressed in the form $\pm \frac{p}{q}$, where $p$ is a factor of the constant term $a_0$ and $q$ is a factor of the leading coefficient $a_n$. For the given function, $a_0 = 8$ and $a_n = -2$. The factors of 8 are $\pm 1, \pm 2, \pm 4, \pm 8$, and the factors of 2 are $\pm 1, \pm 2$. Therefore, the possible rational zeros are $\pm 1, \pm 2, \pm 4, \pm 8$ and their divisions by the factors of the leading coefficient, which only includes division by 2, resulting in $\pm \frac{1}{2}, \pm 1, \pm 2, \pm 4, \pm 8$.

Use the Rational Zero Theorem to List All Possible Rational f(x)=−2x3+2x2−4x+8f ( x ) = - 2 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 8f(x)=−2x3+2x2−4x+8

Use the Rational Zero Theorem to List All Possible Rational $f ( x ) = - 2 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 8$