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The question asks for the \( f \)-intercept and \( x \)-intercepts of the given function:

\[
f(x) = 3x^4 - 24x^3 + 45x^2
\]

### 1. Finding the \( f \)-intercept (y-intercept):
The \( f \)-intercept is the value of the function when \( x = 0 \). 

Substituting \( x = 0 \) into the function:
\[
f(0) = 3(0)^4 - 24(0)^3 + 45(0)^2 = 0
\]
Thus, the \( f \)-intercept is \( 0 \).

### 2. Finding the \( x \)-intercepts:
The \( x \)-intercepts are the points where \( f(x) = 0 \). We need to solve the equation:

\[
3x^4 - 24x^3 + 45x^2 = 0
\]

First, factor out the greatest common factor, which is \( 3x^2 \):
\[
3x^2(x^2 - 8x + 15) = 0
\]

Now, solve for \( x \) by setting each factor equal to zero.

- From \( 3x^2 = 0 \), we get \( x = 0 \).

- Next, solve \( x^2 - 8x + 15 = 0 \). Use the quadratic formula:
\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(15)}}{2(1)}
\]
\[
x = \frac{8 \pm \sqrt{64 - 60}}{2}
\]
\[
x = \frac{8 \pm \sqrt{4}}{2}
\]
\[
x = \frac{8 \pm 2}{2}
\]
Thus, \( x = \frac{8 + 2}{2} = 5 \) and \( x = \frac{8 - 2}{2} = 3 \).

### Final Answers:
- \( f \)-intercept: \( 0 \)
- \( x \)-intercepts: \( x = 0, 3, 5 \)

Would you like further details or have any questions?

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Here are 5 related questions you might explore:
1. How does factoring simplify polynomial equations?
2. What does the multiplicity of an intercept tell us about the graph at that point?
3. How do you use the quadratic formula to solve for roots in more complex cases?
4. What role does the degree of a polynomial play in determining the number of intercepts?
5. How can synthetic division be used to find roots of higher-degree polynomials?

**Tip:** Always check for common factors first when solving polynomial equations. This simplifies the equation and makes finding solutions easier.

Given the function f(x) = 3x^4 - 24x^3 + 45x^2, find its f-intercept and x-intercepts.

This problem demonstrates how to find the f-intercept and x-intercepts of the polynomial function f(x) = 3x^4 - 24x^3 + 45x^2. Learn to use factoring and the quadratic formula to solve for intercepts. Suitable for high school students in Grades 10-12.

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