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We are asked to decompose the function \( f(x) = \frac{18x}{(x-1)(x^2 + x + 1)} \) into partial fractions.

The form of the partial fraction decomposition for this rational function will be:

\[
\frac{18x}{(x-1)(x^2 + x + 1)} = \frac{A}{x-1} + \frac{Bx + C}{x^2 + x + 1}
\]

We need to find the constants \(A\), \(B\), and \(C\) by equating both sides of the equation. Here's the general step-by-step procedure:

### Step 1: Multiply through by the denominator
Multiply both sides by the denominator \((x - 1)(x^2 + x + 1)\) to eliminate the denominator:

\[
18x = A(x^2 + x + 1) + (Bx + C)(x - 1)
\]

### Step 2: Expand both sides
Expand both terms on the right-hand side:

- First term: \( A(x^2 + x + 1) = A x^2 + A x + A \)
- Second term: \((Bx + C)(x - 1) = Bx(x - 1) + C(x - 1) = Bx^2 - Bx + Cx - C \)

So, combining the expanded terms:

\[
18x = A x^2 + A x + A + B x^2 - B x + C x - C
\]

### Step 3: Combine like terms
Group the terms by powers of \(x\):

\[
18x = (A + B)x^2 + (A - B + C)x + (A - C)
\]

### Step 4: Equate coefficients
Now equate the coefficients of \(x^2\), \(x\), and the constant terms from both sides. Since the left-hand side is \(18x\), we compare the coefficients:

1. For \(x^2\): \( A + B = 0 \)
2. For \(x\): \( A - B + C = 18 \)
3. For the constant term: \( A - C = 0 \)

### Step 5: Solve the system of equations
From these, we solve for \(A\), \(B\), and \(C\):

1. \( A + B = 0 \) implies \( B = -A \)
2. \( A - C = 0 \) implies \( A = C \)
3. Substitute into \( A - B + C = 18 \): \( A - (-A) + A = 18 \), which simplifies to \( 3A = 18 \), so \( A = 6 \)

Thus, \( B = -6 \) and \( C = 6 \).

### Final Answer:
The partial fraction decomposition is:

\[
f(x) = \frac{6}{x-1} + \frac{-6x + 6}{x^2 + x + 1}
\]

Would you like further details or explanations on any part of this? Here are 5 related questions to explore:

1. How can partial fraction decomposition help in solving integrals?
2. What is the general method for decomposing higher degree polynomials?
3. How would this approach change if the denominator had repeated roots?
4. Can you apply partial fractions to improper fractions, and how?
5. What are the applications of partial fraction decomposition in differential equations?

**Tip:** Always ensure the degree of the numerator is less than the degree of the denominator before applying partial fractions. If not, perform polynomial division first!

Rewrite the expression f(x) = 18x / ((x-1)(x^2 + x + 1)) using partial fractions.

Learn how to decompose the rational function f(x) = 18x / ((x-1)(x^2 + x + 1)) using partial fractions. This solution includes a step-by-step breakdown of the process, explaining how to find constants and simplify the expression. Ideal for high school or college-level algebra students.

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