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 suggests that any rational root, expressed as a fraction $\frac{p}{q}$, must have $p$ as a factor of the constant term (-400) and $q$ as a factor of the leading coefficient (3). Testing the possible rational roots can be tedious, but for this question, the correct roots are $x = 4, 5, -5, \frac{4}{3}$, which matches option C. This can be verified by either synthetic division, polynomial long division, or using a graphing calculator to find the zeros of the polynomial.

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 $f ( x ) = 3 x ^ { 4 } - 16 x ^ { 3 } - 59 x ^ { 2 } + 400 x - 400$