Find only the rational zeros.
-$$f ( x ) = x ^ { 5 } - 5 x ^ { 4 } + 5 x ^ { 3 } + 15 x ^ { 2 } - 36 x + 20$$
A) $- 1,2 , - 2$
B) $1,2,3$
C) $1,2 , - 2$
D) No rational zeros

Question

Quizplus · Accepted Answer

The rational zeros of a polynomial can be found using the Rational Root Theorem, which states that any rational zero, expressed as a fraction $\frac{p}{q}$, must have $p$ as a factor of the constant term and $q$ as a factor of the leading coefficient. For the given polynomial, the constant term is 20 and the leading coefficient is 1. The factors of 20 are $\pm1, \pm2, \pm4, \pm5, \pm10, \pm20$, and since the leading coefficient is 1, the possible rational zeros are these factors themselves. Testing these values or using synthetic division, we find that $1, 2, \text{ and } -2$ are indeed zeros of the polynomial, making option C correct.

Find Only the Rational Zeros f(x)=x5−5x4+5x3+15x2−36x+20f ( x ) = x ^ { 5 } - 5 x ^ { 4 } + 5 x ^ { 3 } + 15 x ^ { 2 } - 36 x + 20f(x)=x5−5x4+5x3+15x2−36x+20

Find Only the Rational Zeros $f ( x ) = x ^ { 5 } - 5 x ^ { 4 } + 5 x ^ { 3 } + 15 x ^ { 2 } - 36 x + 20$