The factors of `x^3-7x+6` are

Options

A. `x(x-6)(x-1)`
B. `(x^2-6)(x-1)`
C. `(x+1)(x+2)(x-3)`
D. `(x-1)(x+3)\ (x-2)`

Explanation

To factor the expression \( x^3 - 7x + 6 \), we can follow these steps:

### Step 1: Identify the polynomial
We start with the polynomial:
\[ x^3 - 7x + 6 \]

### Step 2: Use the Rational Root Theorem
We can look for rational roots by testing possible factors of the constant term (6) over the leading coefficient (1). The possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 6 \).

### Step 3: Test possible roots
Let's test \( x = 1 \):
\[ 1^3 - 7(1) + 6 = 1 - 7 + 6 = 0 \]
Since \( x = 1 \) is a root, we can factor \( x - 1 \) out of the polynomial.

### Step 4: Perform polynomial long division
Now we will divide \( x^3 - 7x + 6 \) by \( x - 1 \).

1. Divide the leading term: \( x^3 \div x = x^2 \).
2. Multiply \( x^2 \) by \( x - 1 \): \( x^3 - x^2 \).
3. Subtract:
\[ (x^3 - 7x + 6) - (x^3 - x^2) = x^2 - 7x + 6 \]
4. Repeat the process: Divide \( x^2 \) by \( x \) to get \( x \).
5. Multiply \( x \) by \( x - 1 \): \( x^2 - x \).
6. Subtract:
\[ (x^2 - 7x + 6) - (x^2 - x) = -6x + 6 \]
7. Divide \( -6x \) by \( x \) to get \( -6 \).
8. Multiply \( -6 \) by \( x - 1 \): \( -6x + 6 \).
9. Subtract:
\[ (-6x + 6) - (-6x + 6) = 0 \]

The result of the division is:
\[ x^2 + x - 6 \]

### Step 5: Factor the quadratic
Now we need to factor \( x^2 + x - 6 \). We look for two numbers that multiply to \(-6\) (the constant term) and add to \(1\) (the coefficient of \(x\)). The numbers \(3\) and \(-2\) work because:
\[ 3 \times (-2) = -6 \quad \text{and} \quad 3 + (-2) = 1 \]
Thus, we can write:
\[ x^2 + x - 6 = (x + 3)(x - 2) \]

### Step 6: Combine factors
Now we can write the complete factorization of the original polynomial:
\[ x^3 - 7x + 6 = (x - 1)(x + 3)(x - 2) \]

### Final Answer
The factors of \( x^3 - 7x + 6 \) are:
\[ (x - 1)(x + 3)(x - 2) \]

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