To factor the cubic polynomial $x^3 - 2x^2 - x + 2$ using grouping, follow these steps:

Step 1: Group terms

We group the terms of the polynomial into two pairs:

$(x^3 - 2x^2) - (x - 2)$

Step 2: Factor out the greatest common factor (GCF) from each group

Now, factor the greatest common factor from each group:

$x^2(x - 2) - 1(x - 2)$

Step 3: Factor out the common binomial factor

Notice that both terms contain the common binomial factor $(x - 2)$. Factor it out:

$(x - 2)(x^2 - 1)$

Step 4: Factor the difference of squares

Now, observe that $x^2 - 1$ is a difference of squares, which can be factored further:

$x^2 - 1 = (x - 1)(x + 1)$

So, the complete factorization of the polynomial is:

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

Final answer:

The fully factored form of the cubic polynomial $x^3 - 2x^2 - x + 2$ is:

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

Would you like any further details or explanations?

Related questions:

1. How do you factor a cubic polynomial with four terms in general?
2. Can you explain how the difference of squares works?
3. How do you determine if a cubic polynomial is factorable by grouping?
4. What are some other techniques to factor cubic polynomials?
5. Can factoring cubic polynomials help in solving equations?

Tip:

Always check for common factors before attempting to factor by grouping or other methods!