Problem 13 - College Algebra

Question

Factor by grouping. $$ x^{3}-x^{2}+2 x-2 $$

Step-by-Step Solution

Answer

The factored form of the given polynomial \( x^{3}-x^{2}+2x-2 \) by grouping is \( (x^{2}+2)(x-1) \).

Step 1: Group the Terms

In this step, group the polynomial into two. Here, the given polynomial is \( x^{3}-x^{2}+2x-2 \). Group the first two terms together and the last two terms together, which makes \( (x^{3}-x^{2}) + (2x-2) \).

Step 2: Factor out the Greatest Common Factor from Each Group

Find the greatest common factor of each group and factor it out. For the first group, \( x^{3}-x^{2} \), the greatest common factor is \( x^{2} \). For the second group, \( 2x-2 \), the common factor is 2. Factoring these out gives \( x^{2}(x-1) + 2(x-1) \).

Step 3: Factor out the Common Binomial

After factoring out the greatest common coefficients, there is a common binomial term, \( x-1 \). Factoring this out gives \( (x^{2}+2)(x-1) \).

Key Concepts

Understanding Polynomials

A polynomial is a mathematical expression that consists of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. Polynomials are a foundational concept in algebra and can take on many forms. For example, in our exercise, we have the polynomial \( x^3 - x^2 + 2x - 2 \).
This expression is made up of four terms: \( x^3 \), \( -x^2 \), \( 2x \), and \( -2 \). Each term is made up of a coefficient (which can be positive, negative, or zero) and a variable raised to an exponent. The highest exponent of the variable in the polynomial determines its degree; here, the degree is 3 because of the term \( x^3 \). Polynomials can be added, subtracted, and multiplied to form new polynomials. Factoring them involves breaking them down into simpler, multipliable expressions.

Finding the Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides all the terms in a polynomial. It simplifies the polynomial and makes it easier to factor further. Before you factor by grouping, you should always look for a GCF in each group of terms.
In our exercise, we need to find the GCF for the groups \( (x^3 - x^2) \) and \( (2x - 2) \).

• For \( (x^3 - x^2) \), the GCF is \( x^2 \) because it is the largest power of \( x \) that divides both terms.
• For \( (2x - 2) \), the GCF is 2 since it is the largest number that divides both terms.

By factoring out these GCFs, you simplify the polynomial, making it easier to see other common factors.

Using Binomial Factoring

Binomial factoring is a method used to simplify and solve expressions where a polynomial is expressed as two binomials (expressions with two terms). After identifying the GCF in each group, you will often encounter a common binomial factor.
For instance, in the expression \( x^2(x-1) + 2(x-1) \), both terms share the binomial \( (x-1) \) as a factor. When these groups share a common binomial, you can factor it out, simplifying the expression further. Binomial factoring helps transform a more complex polynomial into a simpler product of binomials.

Grouping Terms for Factoring

Grouping terms is a technique used in polynomial factoring, especially effective for polynomials with four or more terms. The goal is to rearrange and group terms so that each group shares a common factor.
In our given exercise, we begin by rearranging \( x^3 - x^2 + 2x - 2 \) as \( (x^3 - x^2) + (2x - 2) \). By grouping the first two terms and the last two terms separately, it becomes easier to identify and factor out the GCF for each group. This method simplifies the factoring process and reveals hidden common factors, paving the way for complete factorization using the binomial factors discovered between groups.