Problem 13 - College Algebra

Question

In Exercises \(11-16,\) factor by grouping. $$x^{3}-x^{2}+2 x-2$$

Step-by-Step Solution

Answer

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

Step 1: Grouping the terms

The given expression is \(x^{3}-x^{2}+2 x-2\). This can be grouped into two separate parts as follows \(x^{3}-x^{2}\) and \(+2 x-2\).

Step 2: Factoring out the GCF from each group

From the first group \(x^{3}-x^{2}\), the GCF is \(x^{2}\), factoring out this, we get \(x^2(x -1)\). From the second group \(2x - 2\), the GCF is 2, factoring out this, we get \(2(x -1)\).

Step 3: Factoring by grouping

Finally, observe that the terms we ended up after factoring the GCF from each group, \(x^2(x -1)\) and \(2(x -1)\), contain a common factor of \(x -1\). Factor out this common factor, you obtain the final factored form, \((x - 1)(x^2 + 2)\).

Key Concepts

Greatest Common Factor (GCF)

When dealing with algebraic expressions, finding the Greatest Common Factor (GCF) is an essential step in simplifying and factoring. The GCF is the largest quantity that divides two or more terms without leaving a remainder. This concept is akin to finding the GCF in basic arithmetic but applied to algebraic terms.

**How to Find the GCF:**

• Identify each term's prime factors, including variables with their lowest exponents.
• For each group, pick the lowest power for each common variable or coefficient.
• The GCF is the product of these selected factors.

In the expression from our example, for \(x^3 - x^2\), the GCF is \(x^2\) as it is the highest power of \(x\) common to both terms. Similarly, in \(2x - 2\), the number 2 is the GCF.

Polynomials

Polynomials are expressions involving variables raised to whole number powers and coefficients. They're a fundamental concept in algebra and can take many forms depending on the number of terms they contain.

**Components of Polynomials:**

• **Monomial:** A single term (e.g., \(7x\)).
• **Binomial:** Two terms (e.g., \(3x + 2\)).
• **Trinomial:** Three terms (e.g., \(x^2 + 5x + 6\)).

Our primary example, \(x^3 - x^2 + 2x - 2\), is a polynomial with four terms. Each term has its own degree based on the exponent of the variable.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition, subtraction, etc.). They form the backbone of algebra and are used to describe relationships and solve problems.

**Understanding Algebraic Expressions:**

• Composed of terms, which can include constants, variables, or both.
• Operations such as addition, subtraction, multiplication, and division combine the terms.
• Expressions can be simplified or factored to solve equations or find specific values.

In the exercise, we started with the expression \(x^3 - x^2 + 2x - 2\), showcasing the use of these basic operations and variables in forming complex expressions. Factoring by grouping was the method used to simplify and transform this expression into a product of simpler ones.