The function f is given by f(x) = 5x³-9x 4/x²-3x 2. At which of the following values of x does the graph of f have an x-intercept?
A) x = 0
B) x = 3
C) x = 5
D) x = 1

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.

The function f is given by f(x) = 5x³-9x + 4/x²-3x + 2. At which of the following values of x does the graph of f have an x-intercept?
A) x = 0
B) x = 3
C) x = 5
D) x = 1

Learn how to find the x-intercept of the rational function f(x) = (5x³ - 9x + 4)/(x² - 3x + 2). This step-by-step solution demonstrates setting the numerator equal to zero and solving for x. Appropriate for students in Grades 10-12 studying algebra and rational functions.

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