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The given function is:

\[
f(x) = 4x^4 - 9x^3 - 36x^2 + 81x
\]

To find the x-intercepts, we need to solve \( f(x) = 0 \). We will factor the function.

### Step 1: Factor out the greatest common factor (GCF).
The GCF of all the terms is \( x \), so we factor out \( x \) from the expression:

\[
f(x) = x(4x^3 - 9x^2 - 36x + 81)
\]

### Step 2: Factor the cubic expression.
Now, we need to factor the cubic polynomial \( 4x^3 - 9x^2 - 36x + 81 \).

We can apply factoring by grouping:

1. Group the terms in pairs:
\[
4x^3 - 9x^2 - 36x + 81 = (4x^3 - 9x^2) - (36x - 81)
\]

2. Factor each group:
\[
= x^2(4x - 9) - 9(4x - 9)
\]

3. Factor out the common binomial factor \( (4x - 9) \):
\[
= (x^2 - 9)(4x - 9)
\]

### Step 3: Factor further if possible.
The expression \( x^2 - 9 \) is a difference of squares, so it can be factored as:
\[
x^2 - 9 = (x - 3)(x + 3)
\]

Thus, the factored form of \( f(x) \) is:
\[
f(x) = x(x - 3)(x + 3)(4x - 9)
\]

### Step 4: Solve for the x-intercepts.
To find the x-intercepts, set each factor equal to zero:
\[
x = 0, \quad x - 3 = 0 \quad \Rightarrow \quad x = 3,
\]
\[
x + 3 = 0 \quad \Rightarrow \quad x = -3,
\]
\[
4x - 9 = 0 \quad \Rightarrow \quad x = \frac{9}{4}.
\]

### Final Answer:
The x-intercepts are \( x = 0 \), \( x = 3 \), \( x = -3 \), and \( x = \frac{9}{4} \).

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How do you find the x-intercepts of a function in general?
2. Can the grouping method be applied to all cubic polynomials?
3. How do you recognize the difference of squares?
4. What are some other methods to factor polynomials?
5. How do you interpret the multiplicity of x-intercepts?

**Tip:** Always check for a greatest common factor (GCF) before applying other factoring techniques.

Factor to find all x-intercepts of the function f(x) = 4x^4 - 9x^3 - 36x^2 + 81x.

Discover how to factor the polynomial f(x) = 4x^4 - 9x^3 - 36x^2 + 81x and find all x-intercepts using factoring by grouping and difference of squares. This detailed explanation is suitable for students in Grades 10-12.

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