Find all rational zeros and factor f(x).
-$$f ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } - x + 3$$
A) $\frac { 1 } { 2 } , - \frac { 1 } { 2 } , 3 ; f ( x ) = ( 2 x - 1 ) ( 2 x + 1 ) ( x - 3 )$
B) $\frac { 1 } { 2 } , - \frac { 1 } { 2 } , 3 ; f ( x ) = ( x - 2 ) ( x + 2 ) ( x - 3 )$
C) $2 , - 2,3 ; f ( x ) = ( x - 2 ) ( x + 2 ) ( x - 3 )$
D) $1 , - 1,3 ; f ( x ) = ( 2 x - 1 ) ( 2 x + 1 ) ( x - 3 )$

Question

Quizplus · Accepted Answer

To find the rational zeros of the polynomial $f(x) = 4x^3 - 12x^2 - x + 3$, we can use the Rational Root Theorem. The possible rational roots are factors of the constant term (3) divided by factors of the leading coefficient (4), which gives us $\pm1, \pm\frac{1}{2}, \pm3, \pm\frac{3}{2}$. Testing these values, we find that $\frac{1}{2}, -\frac{1}{2},$ and $3$ are zeros of $f(x)$. Factoring $f(x)$ using these zeros gives us $f(x) = (2x - 1)(2x + 1)(x - 3)$, which matches option A.

Find All Rational Zeros and Factor F(x) f(x)=4x3−12x2−x+3f ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } - x + 3f(x)=4x3−12x2−x+3

Find All Rational Zeros and Factor F(x) $f ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } - x + 3$