To find the x-intercepts of a function, we need to set the function equal to zero and solve for $x$. The x-intercept occurs when $f(x) = 0$.

You have the function:

$f(x) = \frac{5x^3 - 9x + 4}{x^2 - 3x + 2}$

Step 1: Set $f(x) = 0$

$\frac{5x^3 - 9x + 4}{x^2 - 3x + 2} = 0$ For a fraction to be equal to zero, the numerator must be zero (since the denominator cannot be zero, as this would make the function undefined). So we only need to solve:

$5x^3 - 9x + 4 = 0$

Step 2: Solve $5x^3 - 9x + 4 = 0$

To determine which of the given values of $x$ is a solution, we can substitute each into this equation.

• For $x = 0$: $5(0)^3 - 9(0) + 4 = 0 + 0 + 4 = 4$ So $f(0) = 4$, not zero. Thus, $x = 0$ is not an x-intercept.

• For $x = 3$: $5(3)^3 - 9(3) + 4 = 5(27) - 27 + 4 = 135 - 27 + 4 = 112$ So $f(3) = 112$, not zero. Thus, $x = 3$ is not an x-intercept.

• For $x = 5$: $5(5)^3 - 9(5) + 4 = 5(125) - 45 + 4 = 625 - 45 + 4 = 584$ So $f(5) = 584$, not zero. Thus, $x = 5$ is not an x-intercept.

• For $x = 1$: $5(1)^3 - 9(1) + 4 = 5(1) - 9 + 4 = 5 - 9 + 4 = 0$ So $f(1) = 0$. Therefore, $x = 1$ is an x-intercept.

Final Answer:

The graph of $f(x)$ has an x-intercept at $\boxed{x = 1}$.

Do you want further details on solving cubic equations or simplifying functions?

Here are 5 related questions to explore:

1. What are the other intercepts of this function (y-intercept)?
2. How can we determine the vertical asymptotes for rational functions?
3. What is the process for factoring cubic equations like $5x^3 - 9x + 4 = 0$?
4. How do you find critical points and analyze the graph of rational functions?
5. What are the applications of x-intercepts in real-world scenarios?

Tip: When solving rational functions, focus on the numerator for x-intercepts and check for undefined points using the denominator.