Resolve the following into factor.
`x^(3) - 6x^(2) + 11 x - 6`

Answer

To factor the polynomial \( f(x) = x^3 - 6x^2 + 11x - 6 \), we can follow these steps:

### Step 1: Identify the polynomial
Let \( f(x) = x^3 - 6x^2 + 11x - 6 \).

### Step 2: Use the Rational Root Theorem
We will test possible rational roots. The possible rational roots can be the factors of the constant term (-6) divided by the factors of 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 \( x = 1 \):
\[ f(1) = 1^3 - 6(1^2) + 11(1) - 6 = 1 - 6 + 11 - 6 = 0 \]
Since \( f(1) = 0 \), \( x - 1 \) is a factor of \( f(x) \).

### Step 4: Perform polynomial long division
Now, we will divide \( f(x) \) by \( x - 1 \).

1. Divide \( x^3 \) by \( x \) to get \( x^2 \).
2. Multiply \( x^2 \) by \( x - 1 \) to get \( x^3 - x^2 \).
3. Subtract \( (x^3 - x^2) \) from \( f(x) \):
\[ (x^3 - 6x^2 + 11x - 6) - (x^3 - x^2) = -5x^2 + 11x - 6 \]
4. Divide \( -5x^2 \) by \( x \) to get \( -5x \).
5. Multiply \( -5x \) by \( x - 1 \) to get \( -5x^2 + 5x \).
6. Subtract \( (-5x^2 + 5x) \) from \( -5x^2 + 11x - 6 \):
\[ (-5x^2 + 11x - 6) - (-5x^2 + 5x) = 6x - 6 \]
7. Divide \( 6x \) by \( x \) to get \( 6 \).
8. Multiply \( 6 \) by \( x - 1 \) to get \( 6x - 6 \).
9. Subtract \( (6x - 6) \) from \( 6x - 6 \):
\[ (6x - 6) - (6x - 6) = 0 \]

The result of the division is \( x^2 - 5x + 6 \).

### Step 5: Factor the quadratic
Now we need to factor \( x^2 - 5x + 6 \):
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 6: Combine the factors
Thus, we can write:
\[ f(x) = (x - 1)(x - 2)(x - 3) \]

### Final Answer
The factors of the polynomial \( x^3 - 6x^2 + 11x - 6 \) are:
\[ (x - 1)(x - 2)(x - 3) \]