College Algebra, Chapter 4, 4.4, Section 4.4, Problem 12

A polynomial $P(x) = 3x^3 + 4x^2 - x - 2$ and its graph are given.







a.) List all possible rational zeros of $P$ given by the rational zeros theorem.

possible rational theorem of $\displaystyle P = \frac{\text{factor of 2}}{\text{factor of 3}}$

The factors of $2$ are $\pm 1, \pm 2$ and the factors of $3$ are $\pm 1, \pm 3$. Thus, the possible rational zeros of $P$ are

$\displaystyle \pm \frac{1}{1}, \pm \frac{2}{1}, \pm \frac{1}{3}, \pm \frac{2}{3} $

Simplify the fractions, we get

$\displaystyle \pm 1, \pm 2, \pm \frac{1}{3}, \pm \frac{2}{3}$

b.) From the graph, determine which of the possible rational zeros actually turn out to be zeros.

Based from the graph, the zeros of $P$ are $-1$ and $\displaystyle \frac{2}{3}$.