solve with formula and solution (-18a+3a³+9+6a²) ÷ (-3+3a) synthetic division

To solve \((-18a + 3a^3 + 9 + 6a^2) \div (-3 + 3a)\) using synthetic division, we'll follow these steps:

### Step 1: Arrange the Polynomial
First, arrange the polynomial in descending order of powers of \(a\):

\[
3a^3 + 6a^2 - 18a + 9
\]

### Step 2: Factor the Divisor
The divisor \(-3 + 3a\) can be factored as \(3(a - 1)\). We can ignore the constant factor \(3\) in synthetic division and focus on dividing by \(a - 1\).

### Step 3: Set Up the Synthetic Division
The synthetic division will use the root \(a = 1\) from the divisor \(a - 1\).

- Coefficients of the polynomial \(3a^3 + 6a^2 - 18a + 9\) are \( [3, 6, -18, 9] \).
- The divisor's root is \(a = 1\).

### Step 4: Perform the Synthetic Division

1. **Bring down the first coefficient (3):**
   \[
   3
   \]

2. **Multiply the root (1) by the coefficient (3) and add it to the next coefficient (6):**
   \[
   1 \times 3 = 3
   \]
   \[
   6 + 3 = 9
   \]

3. **Multiply the root (1) by the result (9) and add it to the next coefficient (-18):**
   \[
   1 \times 9 = 9
   \]
   \[
   -18 + 9 = -9
   \]

4. **Multiply the root (1) by the result (-9) and add it to the next coefficient (9):**
   \[
   1 \times -9 = -9
   \]
   \[
   9 - 9 = 0
   \]

The synthetic division table would look like this:

\[
\begin{array}{r|rrrr}
1 & 3 & 6 & -18 & 9 \\
  &   & 3 & 9 & -9 \\
\hline
  & 3 & 9 & -9 & 0 \\
\end{array}
\]

### Step 5: Write Down the Quotient and Remainder

The result from the synthetic division is:

\[
3a^2 + 9a - 9
\]

Since the remainder is 0, the quotient is \(3a^2 + 9a - 9\).

### Final Answer:
\[
\boxed{3a^2 + 9a - 9}
\]

This is the quotient when \((3a^3 + 6a^2 - 18a + 9)\) is divided by \((-3 + 3a)\).

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Do you want more details or have any questions? 

Here are 5 related questions:

1. How do you perform synthetic division when there is a remainder?
2. What are the steps to manually check the result of synthetic division?
3. How can synthetic division be used to factor a polynomial?
4. What is the difference between synthetic and long division in polynomials?
5. How can you use synthetic division to find the roots of a polynomial?

**Tip:** When performing synthetic division, always make sure the polynomial is in standard form with all terms in descending order of the variable's exponent.

Discover how to solve the polynomial division (-18a + 3a³ + 9 + 6a²) ÷ (-3 + 3a) using synthetic division. This detailed step-by-step solution covers key algebraic concepts and is ideal for high school students in Grades 10-12.

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