Use synthetic division to find the quotient and remainder when 6x^6-9x^4+3x^2+9 is divided by x-2.

To use synthetic division to divide \(6x^6 - 9x^4 + 3x^2 + 9\) by \(x - 2\), follow these steps:

### Step 1: Set up the synthetic division

- **Divisor:** \(x - 2\) has a root \(r = 2\).
- **Coefficients:** The polynomial is \(6x^6 - 9x^4 + 3x^2 + 9\). Since the polynomial is missing some terms (for \(x^5\), \(x^3\), and \(x\)), we include coefficients of 0 for these missing terms. The coefficients are:
  \[
  6 \quad 0 \quad -9 \quad 0 \quad 3 \quad 0 \quad 9
  \]

### Step 2: Perform the synthetic division

Place the root \(r = 2\) on the left, and the coefficients on the right. Perform the synthetic division as follows:

\[
\begin{array}{r|rrrrrrr}
2 & 6 & 0 & -9 & 0 & 3 & 0 & 9 \\
  &   & 12 & 24 & 30 & 60 & 126 & 252 \\
\hline
  & 6 & 12 & 15 & 30 & 63 & 126 & 261 \\
\end{array}
\]

### Step 3: Interpret the results

The bottom row represents the coefficients of the quotient and the remainder:

- **Quotient:** The quotient corresponds to \(6x^5 + 12x^4 + 15x^3 + 30x^2 + 63x + 126\).
- **Remainder:** The remainder is \(261\).

### Final Answer:
The quotient is \(6x^5 + 12x^4 + 15x^3 + 30x^2 + 63x + 126\), and the remainder is \(261\).

Would you like a more detailed explanation of any of these steps?

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Here are some related questions for practice:
1. Use synthetic division to divide \(5x^4 - 7x^3 + 2x + 10\) by \(x + 1\).
2. Find the quotient and remainder when \(4x^5 - x^2 + 8x - 7\) is divided by \(x - 3\).
3. Verify the result of the division by multiplying the quotient by the divisor and adding the remainder.
4. Perform synthetic division on \(7x^3 + 4x^2 - x + 2\) divided by \(x - 5\).
5. Determine if \(x - 2\) is a factor of \(6x^6 - 9x^4 + 3x^2 + 9\) by checking the remainder.

**Tip:** Always remember to include a coefficient of zero for any missing terms when setting up synthetic division.

Learn how to use synthetic division to divide the polynomial 6x^6 - 9x^4 + 3x^2 + 9 by the linear divisor x - 2. Detailed step-by-step solution provided, including the quotient and remainder. Suitable for advanced high school students.

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