Problem 16 - Elementary and Intermediate Algebra

Question

Complete each factorization. $$ \begin{aligned} x^{3}-x^{2}+2 x-2 &=(x-1)+(x-1) \\ &=(\quad)\left(x^{2}+2\right) \end{aligned} $$

Step-by-Step Solution

Answer

The complete factorization is \((x-1)(x^2 + 2)\).

Step 1: Identify Grouping

Group the polynomial terms in a way that allows for easy factoring. The expression given is \(x^3 - x^2 + 2x - 2 = (x-1) + (x-1)\). Notice that we can group it into two binomials: \((x^3 - x^2) + (2x - 2)\).

Step 2: Factor Each Group

Factor out the greatest common factor from each group. For \(x^3 - x^2\), you can factor out \(x^2\), resulting in \(x^2(x-1)\). For \(2x - 2\), factor out \(2\), resulting in \(2(x-1)\).

Step 3: Combine Factored Terms

Notice that both groups have a common factor of \((x-1)\). Combine them using this common factor: \((x^2 + 2)(x-1)\).

Step 4: Verify the Factorization

Expand \((x^2 + 2)(x-1)\) to ensure it equals the original polynomial. Distributing, we get \(x^3 - x^2 + 2x - 2\), which matches the original expression.

Key Concepts

Grouping Method

The grouping method is a powerful technique used to factor polynomials, especially when dealing with an expression that has four or more terms. The main idea behind this method is to group terms into smaller groups that can each be factored easily. Let's look at the given polynomial:
o One way to begin is by looking for pairs of terms that have common factors. In our exercise, we have the expression \(x^3 - x^2 + 2x - 2\).
o We can group the terms as follows: \((x^3 - x^2) + (2x - 2)\). Notice how we've split the polynomial into two groups.
By using the grouping method, our goal is to create groups that can each be factored separately. This often simplifies the original problem significantly. Just remember to choose groupings where you can identify common factors, making the next step of factoring much easier.

Greatest Common Factor

To factor each group, we need to identify any greatest common factor (GCF) present in the pairs. The greatest common factor is the largest expression that divides each term in the group without leaving a remainder.
In our example:

• For the first group, \(x^3 - x^2\), the GCF is \(x^2\). Factoring this out, we get \(x^2(x-1)\).
  
• In the second group \(2x - 2\), the GCF is \(2\). Factoring out, gives \(2(x-1)\).
  

Identifying and factoring out the GCF is critical as it simplifies each part of the polynomial, making it possible to see common binomial factors. Always check each term to ensure we're factoring out the largest possible GCF. This step helps us to prepare the expression for further simplification into a single product.

Binomial Factorization

After factoring out the greatest common factors from each group, we observe that both groups share a common binomial, which is the key to completing the factorization process. The binomial in this context is \((x-1)\). This commonality allows us to factor it out, allowing us to write the expression as a product of two factors.
The original problem was reduced to:

• Factor from the grouped expression: \(x^2(x-1) + 2(x-1)\)
• Notice the common factor \((x-1)\) in both terms.
  

Pull \((x-1)\) out: \[(x^2 + 2)(x-1)\]
The process of binomial factorization is crucial because it brings everything together. It turns our polynomial into a perfectly factorized expression, making it simpler to work with for further operations or for solving equations. Always check by expanding to verify your factored expression equals the original polynomial.