We are tasked with finding the rational zeros of the polynomial $P(x) = 3x^4 + x^3 - 9x^2 - 9x - 2$, based on the graph and the information provided.

Step 1: Apply the Rational Root Theorem

The Rational Root Theorem suggests that any possible rational zeros must be of the form $\frac{p}{q}$, where:

• $p$ is a factor of the constant term (in this case, $-2$),
• $q$ is a factor of the leading coefficient (in this case, $3$).

Factors of the constant term $-2$: $\pm 1, \pm 2$

Factors of the leading coefficient $3$: $\pm 1, \pm 3$

Thus, the possible rational zeros are: $\pm 1, \pm 2, \pm \frac{1}{3}, \pm \frac{2}{3}$

Step 2: Analyze the Graph for Zeros

From the description of the graph:

• The polynomial crosses the x-axis at approximately $x = -1$, $x = 2$, and possibly at other locations.
• The graph also changes direction at points where the polynomial has turning points or possibly repeated zeros.

Step 3: Identify the Zeros from the Graph

From the description of the graph, the zeros of the polynomial occur at:

• $x = -1$
• $x = 2$

We can conclude that the possible rational zeros that are actually zeros of the polynomial are $x = -1$ and $x = 2$.

Step 4: Determine the Multiplicities

• The point $(-1, 0)$ is where the graph changes direction, suggesting a root with an even multiplicity.
• The point $(2, 0)$ suggests a simple zero, as the graph crosses the x-axis.

Thus, the multiplicities of the zeros are:

• $x = -1$ has multiplicity 2 (even multiplicity because the graph changes direction at this point).
• $x = 2$ has multiplicity 1 (simple zero because the graph crosses the x-axis).

Final Answer:

The zeros are: $x = -1, x = 2$

Would you like to verify this result by factoring the polynomial, or would you like further explanation on any step? Let me know!