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To solve this problem using synthetic division, we need to divide the polynomial \( x^5 + 6x^3 - 7 \) by \( x - 1 \).

### Step-by-Step Solution
1. **Set up the synthetic division:** Since we're dividing by \( x - 1 \), our divisor's root is \( x = 1 \). We will use \( 1 \) for the synthetic division process.

2. **Write the coefficients of the polynomial:** Notice that the polynomial \( x^5 + 6x^3 - 7 \) is missing terms for \( x^4 \), \( x^2 \), and \( x^1 \). We need to represent these as \(0\) in the synthetic division setup.
   - The polynomial \( x^5 + 0x^4 + 6x^3 + 0x^2 + 0x - 7 \) has coefficients: \( 1, 0, 6, 0, 0, -7 \).

3. **Perform synthetic division:**
   - Start with \(1\) outside and the coefficients \(1, 0, 6, 0, 0, -7\) in a row.
   - Bring down the leading coefficient (1).
   - Multiply by \(1\) (the root), add the result to the next coefficient, and repeat this process.

Here is how the division proceeds:

| Coefficients    | 1     | 0     | 6     | 0     | 0     | -7    |
|-----------------|-------|-------|-------|-------|-------|-------|
| Steps           |       | \(1\) |       |       |       |       |
| Synthetic Steps |       | 1     |       |       |       |       |
| Result          |       |       |       |       |       |       |



Divide \( \frac{x^5 + 6x^3 - 7}{x - 1} \) using synthetic division.

Learn how to use synthetic division to divide the polynomial \( x^5 + 6x^3 - 7 \) by \( x - 1 \). This problem guides you through the steps of synthetic division and is ideal for high school algebra students in Grades 10-12.

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