https://file.aiquickdraw.com/mathbot/file/1730247692127iacrfpy2.jpg

To solve this using synthetic division, we follow these steps:

The polynomial is:
\[
7x^6 - 2x^4 + 8x^2 + 9
\]
and we need to divide by \( x - 2 \). In synthetic division, we use the root of \( x - 2 = 0 \), which is \( x = 2 \).

1. **Write down the coefficients** of the polynomial, filling in zeros for any missing terms:
   \[
   7, 0, -2, 0, 8, 0, 9
   \]

2. **Set up synthetic division** using \( 2 \) as the divisor:
   
   ```
     2 |  7   0  -2   0   8   0   9
        |      14  28  52  104 224 448
        ---------------------------------
          7  14  26  52  112 224 457
   ```

3. **Perform the division** by following these steps:
   - Bring down the first coefficient, \(7\).
   - Multiply \(7\) by \(2\) to get \(14\), and add to the next coefficient \(0\), resulting in \(14\).
   - Multiply \(14\) by \(2\) to get \(28\), and add to the next coefficient \(-2\), resulting in \(26\).
   - Multiply \(26\) by \(2\) to get \(52\), and add to the next coefficient \(0\), resulting in \(52\).
   - Multiply \(52\) by \(2\) to get \(104\), and add to the next coefficient \(8\), resulting in \(112\).
   - Multiply \(112\) by \(2\) to get \(224\), and add to the next coefficient \(0\), resulting in \(224\).
   - Multiply \(224\) by \(2\) to get \(448\), and add to the last coefficient \(9\), resulting in \(457\).

4. **Interpret the result**:
   - The quotient is: \( 7x^5 + 14x^4 + 26x^3 + 52x^2 + 112x + 224 \)
   - The remainder is: \( 457 \)

### Final Answer:
- Quotient: \( 7x^5 + 14x^4 + 26x^3 + 52x^2 + 112x + 224 \)
- Remainder: \( 457 \)

Would you like a breakdown of any step?

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Here are 5 related questions to expand your understanding:

1. How does synthetic division differ from polynomial long division?
2. What is the significance of the remainder in synthetic division?
3. Can synthetic division be used for divisors other than linear binomials (like \( x - c \))?
4. What does the remainder tell us about whether the divisor is a factor of the polynomial?
5. How would the solution change if the divisor was \( x + 2 \) instead of \( x - 2 \)?

**Tip:** In synthetic division, always remember to fill in zeroes for any missing terms in the polynomial to keep the alignment correct.

Use synthetic division to find the quotient and remainder when 7x^6 - 2x^4 + 8x^2 + 9 is divided by x - 2.

Learn how to use synthetic division to find the quotient and remainder when dividing the polynomial 7x^6 - 2x^4 + 8x^2 + 9 by x - 2. This step-by-step solution covers polynomial division, synthetic division techniques, and application of the remainder theorem. Ideal for high school students in Grades 10-12.

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