One factor of `x ^(3) - 7x + 6 `is `x -1.` The other factors are :

Options

A. `(x-3) (x+2)`
B. `(x+3) (x-2)`
C. `(x-3) (x-2)`
D. `(x+3) (x+2)`

Answer

To find the other factors of the polynomial \( x^3 - 7x + 6 \) given that one factor is \( x - 1 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Polynomial and Given Factor**:
We have the polynomial \( P(x) = x^3 - 7x + 6 \) and one of its factors is \( x - 1 \).

2. **Perform Polynomial Long Division**:
We will divide \( P(x) \) by \( x - 1 \).

- **Divide the leading term**: \( x^3 \div x = x^2 \).
- **Multiply**: \( x^2 \cdot (x - 1) = x^3 - x^2 \).
- **Subtract**:
\[ (x^3 - 7x + 6) - (x^3 - x^2) = x^2 - 7x + 6. \]

3. **Repeat the Division**:
Now we need to divide \( x^2 - 7x + 6 \) by \( x - 1 \).

- **Divide the leading term**: \( x^2 \div x = x \).
- **Multiply**: \( x \cdot (x - 1) = x^2 - x \).
- **Subtract**:
\[ (x^2 - 7x + 6) - (x^2 - x) = -6x + 6. \]

4. **Final Division**:
Now we need to divide \( -6x + 6 \) by \( x - 1 \).

- **Divide the leading term**: \( -6x \div x = -6 \).
- **Multiply**: \( -6 \cdot (x - 1) = -6x + 6 \).
- **Subtract**:
\[ (-6x + 6) - (-6x + 6) = 0. \]

5. **Result of Division**:
The result of the division is \( x^2 + x - 6 \). Thus, we can write:
\[ P(x) = (x - 1)(x^2 + x - 6). \]

6. **Factor the Quadratic**:
Now we need to factor \( x^2 + x - 6 \). We look for two numbers that multiply to \(-6\) and add to \(1\). The numbers \(3\) and \(-2\) work.

- We can write:
\[ x^2 + 3x - 2x - 6 = (x^2 + 3x) + (-2x - 6). \]
- Factor by grouping:
\[ x(x + 3) - 2(x + 3) = (x - 2)(x + 3). \]

7. **Final Factors**:
Thus, the complete factorization of the polynomial is:
\[ P(x) = (x - 1)(x - 2)(x + 3). \]

### Conclusion:
The other factors of \( x^3 - 7x + 6 \) are \( x - 2 \) and \( x + 3 \).