Use grouping to factor the following cubic polynomial.
x
3
−
2
x
2
−
x
+
2

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!

Use grouping to factor the following cubic polynomial: x^3 - 2x^2 - x + 2

Learn how to factor the cubic polynomial x^3 - 2x^2 - x + 2 by grouping. This step-by-step solution covers important algebraic concepts, including the factor theorem, the difference of squares, and polynomial factorization. Suitable for high school students in Grades 10-12.

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