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

Question

Quizplus · Accepted Answer

The rational zeros can be found using the Rational Root Theorem, which suggests that any rational root, in the form of $\frac{p}{q}$, is a factor of the constant term (in this case, 2) divided by a factor of the leading coefficient (in this case, 4). The possible rational roots are $\pm1, \pm2, \pm\frac{1}{2}, \pm\frac{1}{4}$. Testing these, we find that $\frac{1}{2}$, $-\frac{1}{2}$, and $2$ are indeed zeros of the polynomial. Factoring $f(x)$ using these zeros, we get $f(x) = (2x - 1)(2x + 1)(x - 2)$, which corresponds to choice B.

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

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