To solve this problem, we'll use the Rational Zero Test (also known as the Rational Root Theorem) to list the possible rational zeros for the polynomial function:

$f(x) = 2x^4 - 21x^3 + 49x^2 + 9x - 63$

Step 1: The Rational Zero Test

The Rational Zero Test states that any rational zero of a polynomial function must be of the form:

$\frac{p}{q}$

Where:

• $p$ is a factor of the constant term (the term without $x$, which is $-63$).
• $q$ is a factor of the leading coefficient (the coefficient of the highest degree term, which is $2$).

Step 2: Identify factors of the constant term and leading coefficient

• The constant term is $-63$, and its factors are: $p = \pm 1, \pm 3, \pm 7, \pm 9, \pm 21, \pm 63$
• The leading coefficient is $2$, and its factors are: $q = \pm 1, \pm 2$

Step 3: List all possible rational zeros

To list all possible rational zeros, we'll take all combinations of the factors of $p$ divided by the factors of $q$:

$\frac{p}{q} = \pm \frac{1}{1}, \pm \frac{3}{1}, \pm \frac{7}{1}, \pm \frac{9}{1}, \pm \frac{21}{1}, \pm \frac{63}{1}, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{7}{2}, \pm \frac{9}{2}, \pm \frac{21}{2}, \pm \frac{63}{2}$

Thus, the possible rational zeros are:

$\pm 1, \pm 3, \pm 7, \pm 9, \pm 21, \pm 63, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{7}{2}, \pm \frac{9}{2}, \pm \frac{21}{2}, \pm \frac{63}{2}$

Step 4: Verify the zeros shown on the graph

Based on the graph description, the zeros of the function $f(x)$ are:

x = -1, \quad x = 1.5 \quad \text{(or \frac{3}{2})}, \quad x = 3

Now let's check if these values are in our list of possible rational zeros:

• $x = -1$ is in the list.
• $x = \frac{3}{2}$ (which is $1.5$) is in the list.
• $x = 3$ is in the list.

Thus, all zeros of $f(x)$ shown on the graph are contained in our list of possible rational zeros.

Final Answer:

The possible rational zeros of $f(x)$ are:

$\pm 1, \pm 3, \pm 7, \pm 9, \pm 21, \pm 63, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{7}{2}, \pm \frac{9}{2}, \pm \frac{21}{2}, \pm \frac{63}{2}$

Would you like further explanation or details on any step?

Here are five related questions to expand on this topic:

1. How do you apply synthetic division to verify which possible rational zeros are actual zeros?
2. How do you use Descartes' Rule of Signs to predict the number of positive and negative real zeros?
3. What is the significance of the multiplicity of zeros in the graph of a polynomial function?
4. How do you factor a polynomial after finding one rational zero?
5. What is the difference between rational zeros and irrational zeros in terms of solving polynomials?

Tip: The Rational Root Theorem is useful when working with polynomials with integer coefficients, as it gives a manageable list of possible rational solutions.