Question

use the following function to answer parts a through c.
f(x)=x^{3}+5x^{2}-30x - 34

a. list all rational zeros that are possible according to the rational zero theorem.
(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)

Explanation:

Step1: Identify the leading - coefficient and constant term

For the polynomial $f(x)=x^{3}+5x^{2}-30x - 34$, the leading - coefficient $a_{n}=1$ and the constant term $a_{0}=-34$.

Step2: Find the factors of the leading - coefficient and constant term

The factors of the leading - coefficient $a_{n}=1$ are $\pm1$. The factors of the constant term $a_{0}=-34$ are $\pm1,\pm2,\pm17,\pm34$.

Step3: Apply the Rational Zero Theorem

The Rational Zero Theorem states that if a polynomial $f(x)=a_{n}x^{n}+a_{n - 1}x^{n-1}+\cdots+a_{1}x + a_{0}$ has integer coefficients, then the possible rational zeros are of the form $\frac{p}{q}$, where $p$ is a factor of the constant term $a_{0}$ and $q$ is a factor of the leading - coefficient $a_{n}$.
Since $q = \pm1$ and $p=\pm1,\pm2,\pm17,\pm34$, the possible rational zeros are $\pm1,\pm2,\pm17,\pm34$.

Answer:

$1, - 1,2,-2,17,-17,34,-34$