Find all the rational zeros.
-$$f ( x ) = 4 x ^ { 4 } + 9 x ^ { 3 } - 34 x ^ { 2 } - 25 x - 4$$
A) $\frac { 1 } { 4 } , 4 , \pm 1$
B) $- \frac { 1 } { 4 } , - 4$
C) $\frac { 1 } { 4 } , 4$
D) $- \frac { 1 } { 4 } , - 4,1$

Question

Quizplus · Accepted Answer

To find the rational zeros of the polynomial $f(x) = 4x^4 + 9x^3 - 34x^2 - 25x - 4$, 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 and $q$ as a factor of the leading coefficient. Here, the constant term is -4, and the leading coefficient is 4. The factors of -4 are $\pm1, \pm2, \pm4$, and the factors of 4 are $\pm1, \pm2, \pm4$. Therefore, the possible rational roots are $\pm1, \pm2, \pm4, \pm\frac{1}{2}, \pm\frac{1}{4}$. Testing these values or using synthetic division, we find that $-\frac{1}{4}$ and $-4$ are indeed roots of the polynomial, making option B correct.

Find All the Rational Zeros f(x)=4x4+9x3−34x2−25x−4f ( x ) = 4 x ^ { 4 } + 9 x ^ { 3 } - 34 x ^ { 2 } - 25 x - 4f(x)=4x4+9x3−34x2−25x−4

Find All the Rational Zeros $f ( x ) = 4 x ^ { 4 } + 9 x ^ { 3 } - 34 x ^ { 2 } - 25 x - 4$