Find all the rational zeros of the function $f ( x ) = 3 x ^ { 4 } - 16 x ^ { 3 } - 59 x ^ { 2 } + 400 x - 400$.
A) $x = - 4,5 , - 5 , - \frac { 3 } { 4 }$
B) $x = 3 , - 20,5$
C) $x = 4,5 , - 5 , \frac { 4 } { 3 }$
D) $x = - \frac { 4 } { 5 } , \frac { 5 } { 3 } , \frac { 4 } { 3 } , - 5$
E) $x = 3 , - 20 , \frac { 5 } { 3 } , \frac { 4 } { 3 }$

Question

Quizplus · Accepted Answer

To find the rational zeros of the polynomial $f(x) = 3x^4 - 16x^3 - 59x^2 + 400x - 400$, we can use the Rational Root Theorem. This theorem states that any rational root, expressed in its lowest terms as $\frac{p}{q}$, must have $p$ as a factor of the constant term (-400) and $q$ as a factor of the leading coefficient (3). By testing possible rational roots, we find that $x = 4, 5, -5$, and $\frac{4}{3}$ are indeed zeros of the polynomial, making option C correct.

Find All the Rational Zeros of the Function f(x)=3x4−16x3−59x2+400x−400f ( x ) = 3 x ^ { 4 } - 16 x ^ { 3 } - 59 x ^ { 2 } + 400 x - 400f(x)=3x4−16x3−59x2+400x−400

Find All the Rational Zeros of the Function $f ( x ) = 3 x ^ { 4 } - 16 x ^ { 3 } - 59 x ^ { 2 } + 400 x - 400$