Choose the one alternative that best completes the statement or answers the question.
List the possible rational zeros.
-$$f ( x ) = 25 x ^ { 4 } - 7 x ^ { 3 } + 2 x + 10$$
A) $\pm 1 , \pm \frac { 1 } { 5 } , \pm \frac { 1 } { 25 } , \pm 5 , \pm 2 , \pm 10 , \pm \frac { 2 } { 5 } , \pm \frac { 2 } { 25 }$
B) $\pm 1 , \pm \frac { 1 } { 5 } , \pm \frac { 1 } { 25 } , \pm 5 , \pm \frac { 5 } { 2 } , \pm 25 , \pm \frac { 25 } { 2 }$
C) $- 1 , - \frac { 1 } { 5 } , - \frac { 1 } { 25 } , - 5 , - 2 , - \frac { 2 } { 5 } , - \frac { 2 } { 25 }$
D) $1 , \frac { 1 } { 5 } , \frac { 1 } { 25 } , 5,2 , \frac { 2 } { 5 } , \frac { 2 } { 25 }$

Question

Quizplus · Accepted Answer

The possible rational zeros of a polynomial can be found using the Rational Root Theorem, which states that any rational root, expressed in its lowest terms p/q, is such that p is a factor of the constant term and q is a factor of the leading coefficient. For the polynomial $25x^4 - 7x^3 + 2x + 10$, the constant term is 10 and the leading coefficient is 25. The factors of 10 are $\pm1, \pm2, \pm5, \pm10$ and the factors of 25 are $\pm1, \pm5, \pm25$. Therefore, the possible rational zeros include all combinations of the factors of the constant term over the factors of the leading coefficient, leading to the options listed in choice A.

Choose the One Alternative That Best Completes the Statement or Answers