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**Problem**:  
"Perform the following synthetic division and give your final answer in decimal."
\[ (36x^3 - 18x^2 + 21x + 108) \div (3x - 3) \]

Let's perform synthetic division on this expression.

### Step 1: Set up the synthetic division.

We are dividing \( 36x^3 - 18x^2 + 21x + 108 \) by \( 3x - 3 \). First, divide the leading term of the divisor \( 3x - 3 \) by 3, which gives \( x - 1 \).

Thus, we will divide the polynomial \( 36x^3 - 18x^2 + 21x + 108 \) by \( x - 1 \).

### Step 2: Apply synthetic division.

- Coefficients of the polynomial: \( [36, -18, 21, 108] \).
- The zero of the divisor \( x - 1 \) is \( x = 1 \).

Now we perform the synthetic division steps:

1. Bring down the first coefficient \( 36 \).
2. Multiply \( 36 \) by 1 (the divisor root) and add it to the next coefficient:  
   \[ -18 + (36 \times 1) = 18 \]
3. Multiply \( 18 \) by 1 and add to the next coefficient:  
   \[ 21 + (18 \times 1) = 39 \]
4. Multiply \( 39 \) by 1 and add to the next coefficient:  
   \[ 108 + (39 \times 1) = 147 \]

### Step 3: Write the quotient and remainder.

The quotient is \( 36x^2 + 18x + 39 \) with a remainder of 147.

### Final Answer:
\[
36x^2 + 18x + 39 + \frac{147}{3x - 3}
\]

Would you like a more detailed explanation of synthetic division or how this result was obtained?

Here are some related questions for further practice:

1. How do you perform synthetic division when dividing by a binomial like \( x + 2 \)?
2. What are the conditions under which synthetic division can be used?
3. Can you explain how the remainder theorem relates to synthetic division?
4. How does synthetic division differ from polynomial long division?
5. Can synthetic division be applied when dividing by a quadratic?

**Tip**: When performing synthetic division, always check the divisor to see if it can be simplified first, as this will make the division easier.

Perform the following synthetic division and give your final answer in decimal: (36x^3 - 18x^2 + 21x + 108) ÷ (3x - 3).

Learn how to perform synthetic division on the polynomial (36x^3 - 18x^2 + 21x + 108) divided by (3x - 3). This detailed solution explains each step in the synthetic division process and provides the final answer, with an exploration of key algebraic concepts. Suitable for students in Grades 10-12.

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