Find all rational zeros.
-$$f ( x ) = 4 x ^ { 3 } - 8 x ^ { 2 } - x + 2$$
A) $\frac { 1 } { 2 } , - \frac { 1 } { 2 } , 2$
B) $1 , - 1,2$
C) $\frac { 1 } { 2 } , - \frac { 1 } { 2 } , - 2$
D) $2 , - 2,2$

Question

Quizplus · Accepted Answer

The rational root theorem suggests that the rational zeros of a polynomial $f(x) = a_nx^n + \ldots + a_1x + a_0$ can be found among the fractions $\pm \frac{p}{q}$, where $p$ is a factor of the constant term $a_0$ and $q$ is a factor of the leading coefficient $a_n$. For $f(x) = 4x^3 - 8x^2 - x + 2$, the constant term $a_0 = 2$ and the leading coefficient $a_n = 4$. The factors of 2 are $\pm1, \pm2$ and the factors of 4 are $\pm1, \pm2, \pm4$, so the possible rational zeros are $\pm1, \pm\frac{1}{2}, \pm2, \pm\frac{1}{2}$. Testing these values in the polynomial, we find that $\frac{1}{2}, -\frac{1}{2},$ and $2$ are indeed zeros of the polynomial, making option A correct.

Find All Rational Zeros f(x)=4x3−8x2−x+2f ( x ) = 4 x ^ { 3 } - 8 x ^ { 2 } - x + 2f(x)=4x3−8x2−x+2

Find All Rational Zeros $f ( x ) = 4 x ^ { 3 } - 8 x ^ { 2 } - x + 2$