To decompose the given rational function \( \frac{3x+6}{(x+9)^3} \) into partial fractions, we will express it as a sum of simpler fractions with linear denominators.

### Step 1: Set up the form for partial fraction decomposition.
Since the denominator has a repeated linear factor \((x+9)^3\), the decomposition will take the form:

\[
\frac{3x+6}{(x+9)^3} = \frac{A}{x+9} + \frac{B}{(x+9)^2} + \frac{C}{(x+9)^3}
\]

### Step 2: Multiply through by \((x+9)^3\).
Multiply both sides of the equation by the denominator \((x+9)^3\) to eliminate the denominators:

\[
3x+6 = A(x+9)^2 + B(x+9) + C
\]

### Step 3: Expand the right-hand side.
Expand each term on the right-hand side:

- \( A(x+9)^2 = A(x^2 + 18x + 81) \)
- \( B(x+9) = Bx + 9B \)

So, we have:

\[
3x+6 = A(x^2 + 18x + 81) + B(x + 9) + C
\]

Expanding further:

\[
3x+6 = A(x^2) + A(18x) + A(81) + B(x) + 9B + C
\]

\[
3x+6 = A x^2 + (18A + B)x + (81A + 9B + C)
\]

### Step 4: Compare coefficients.
Now, compare the coefficients of \(x^2\), \(x\), and the constant terms on both sides:

- For \(x^2\): \( A = 0 \)
- For \(x\): \( 18A + B = 3 \)
- For the constant: \( 81A + 9B + C = 6 \)

### Step 5: Solve for \(A\), \(B\), and \(C\).
From \( A = 0 \), substitute this into the other two equations:

- \( 18(0) + B = 3 \) → \( B = 3 \)
- \( 81(0) + 9(3) + C = 6 \) → \( 27 + C = 6 \) → \( C = -21 \)

### Step 6: Write the final partial fraction decomposition.
Substitute the values of \(A\), \(B\), and \(C\) into the original partial fraction form:

\[
\frac{3x+6}{(x+9)^3} = \frac{3}{(x+9)^2} - \frac{21}{(x+9)^3}
\]

This is the partial fraction decomposition of \( \frac{3x+6}{(x+9)^3} \).

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Do you have any questions or would you like more details? Here are five related questions to expand this topic:

1. How do you approach partial fraction decomposition with irreducible quadratic factors?
2. What is the general rule for decomposing rational functions with repeated linear factors?
3. Can you use partial fractions to solve definite integrals?
4. How would you handle the decomposition if the numerator were a higher degree than the denominator?
5. How can partial fractions be applied in Laplace transforms?

**Tip:** Always check the degree of the numerator and denominator before attempting partial fraction decomposition. If the degree of the numerator is greater, perform polynomial division first.

Learn how to decompose the rational function 3x+6/(x+9)^3 using partial fraction decomposition. This solution covers the method for repeated linear factors and is suitable for students in Grade 11-12. Detailed step-by-step explanation included.

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