Find all rational zeros.
-$$f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 14 x + 24$$
A) $- 5 , - 7,1$
B) $5,7 , - 1$
C) $- 4 , - 6,1$
D) $4,6 , - 1$

Question

Quizplus · Accepted Answer

To find the rational zeros of the polynomial $f(x) = x^3 - 9x^2 + 14x + 24$, we can use the Rational Root Theorem. This theorem states that any rational root, expressed as a fraction $\frac{p}{q}$ in simplest form, has $p$ as a factor of the constant term (in this case, 24) and $q$ as a factor of the leading coefficient (in this case, 1). Thus, the possible rational roots are the factors of 24, which are $\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12, \pm24$.By testing these possible roots, we find that $x = 4$ and $x = 6$ are indeed roots of the polynomial because $f(4) = 0$ and $f(6) = 0$. To find the third root, we can either use synthetic division or directly test the remaining candidates. Since $f(x)$ is a cubic polynomial and we have found two roots, the third root can be found by dividing the polynomial by $(x-4)(x-6)$ or by testing the remaining factors. Testing $x = -1$ gives $f(-1) = 0$, confirming it as the third root.Therefore, the rational zeros of the polynomial are $4, 6, -1$, which corresponds to choice D.

Find All Rational Zeros f(x)=x3−9x2+14x+24f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 14 x + 24f(x)=x3−9x2+14x+24

Find All Rational Zeros $f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 14 x + 24$