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To solve this, we can use synthetic division, which is often used when dividing polynomials by a binomial of the form \( x - c \).

### Problem
We need to divide:
\[
7x^6 - 2x^4 + 8x^2 + 9
\]
by \( x - 2 \).

Since synthetic division requires every power of \( x \) from the highest degree down to zero, we’ll rewrite the polynomial with coefficients for all powers of \( x \), filling in missing terms with 0.

### Polynomial Coefficients
The polynomial \( 7x^6 - 2x^4 + 8x^2 + 9 \) has coefficients:
\[
[7, 0, -2, 0, 8, 0, 9]
\]
And since \( x - 2 \) means \( c = 2 \), we’ll perform synthetic division using \( 2 \).

### Synthetic Division Steps
1. **Bring down the leading coefficient:** Start with \( 7 \).
2. **Multiply and add:** Multiply \( 7 \) by \( 2 \) and add it to the next coefficient.
3. **Repeat** for each coefficient.

Let’s go through the calculation step-by-step.

#### Synthetic Division Table
| Step         | Coefficients | Calculation          | Result  |
|--------------|--------------|----------------------|---------|
| Start        | 7            |                      | 7       |
| \(7 \times 2\)    | 0            | \(7 \times 2 = 14\)      | 14      |
| \(14 + 0\)   | 0            | \(14 + 0 = 14\)       | 14      |
| \(14 \times 2\)   | -2           | \(14 \times 2 = 28\)     | 28      |
| \(28 - 2\)   | 0            | \(28 - 2 = 26\)       | 26      |
| \(26 \times

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 divide the polynomial 7x^6 - 2x^4 + 8x^2 + 9 by x - 2. This solution includes a step-by-step breakdown, showing how to find the quotient and remainder using synthetic division. Ideal for high school students in Grades 10-12.

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