Determine all possible rational solutions of the polynomial equation.
-$$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 4 x - 24$$
A) $\pm 1 , \pm 2 , \pm 3 , \pm 4 , \pm 6 , \pm 8 , \pm 12 , \pm 24$
B) $\pm 1 , \pm 2 , \pm 3 , \pm 4 , \pm 6 , \pm 12 , \pm 24$
C) $\pm 1 , \pm 2 , \pm 3 , \pm 4 , \pm 24$
D) $\pm 1 , \pm \frac { 1 } { 2 } , \pm 2 , \pm 3 , \pm 4 , \pm 6 , \pm 8 , \pm 12 , \pm 24$

Question

Quizplus · Accepted Answer

The Rational Root Theorem states that any rational solution of the polynomial equation $f(x) = x^3 - 6x^2 + 4x - 24$ can be expressed in the form $\pm \frac{p}{q}$, where $p$ is a factor of the constant term (24) and $q$ is a factor of the leading coefficient (1). Since the leading coefficient is 1, all factors of the constant term are potential rational solutions. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24, thus the possible rational solutions are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24$.

Determine All Possible Rational Solutions of the Polynomial Equation f(x)=x3−6x2+4x−24f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 4 x - 24f(x)=x3−6x2+4x−24

Determine All Possible Rational Solutions of the Polynomial Equation $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 4 x - 24$