Find all rational zeros.
-$$f ( x ) = 10 x ^ { 3 } + 53 x ^ { 2 } + 14 x - 5$$
A) $- 2 , - 5 , - 5$
B) $- 2,5 , - 5$
C) $- \frac { 1 } { 10 } , 1 , - 5$
D) $- \frac { 1 } { 2 } , \frac { 1 } { 5 } , - 5$

Question

Quizplus · Accepted Answer

To find the rational zeros of the polynomial $f(x) = 10x^3 + 53x^2 + 14x - 5$, we 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 (-5) and $q$ as a factor of the leading coefficient (10). The factors of -5 are $\pm1, \pm5$, and the factors of 10 are $\pm1, \pm2, \pm5, \pm10$. Testing these combinations, we find that $-\frac{1}{2}$, $\frac{1}{5}$, and $-5$ are the roots that satisfy the polynomial equation, making option D correct.

Find All Rational Zeros f(x)=10x3+53x2+14x−5f ( x ) = 10 x ^ { 3 } + 53 x ^ { 2 } + 14 x - 5f(x)=10x3+53x2+14x−5

Find All Rational Zeros $f ( x ) = 10 x ^ { 3 } + 53 x ^ { 2 } + 14 x - 5$