Factorize the following expression:
x³ - 2x² - x + 2.

To factor the expression x³ - 2x² - x + 2, we can use the method of grouping. First, we can rearrange the terms for easier grouping:

Step 1: Group the Terms

We can group the first two terms and the last two terms:

• (x³ - 2x²) + (-x + 2)

Step 2: Factor Out Common Factors

Now, we factor out the common factors from each group:

• x²(x - 2) - 1(x - 2)

Step 3: Factor by Grouping

Notice that both groups contain the factor (x - 2). We can factor this out:

• (x - 2)(x² - 1)

Step 4: Factor the Remaining Expression

The expression x² - 1 is a difference of squares, which can be factored further:

• (x - 1)(x + 1)

Final Factored Form

Putting it all together, the complete factorization of the original expression is:

(x - 2)(x - 1)(x + 1)