Problem 14 - College Algebra Essentials

Question

$$\text { Factor by grouping.}$$ $$x^{3}+6 x^{2}-2 x-12$$

Step-by-Step Solution

Answer

The factored form of the expression \(x^{3}+6 x^{2}-2 x-12\) by grouping is \( (x^{2}-2)(x+6)\).

Step 1: Arrange Expression

First, rearrange the expression into two groups. We split the expression into two groups such that there are common factors within each group. This gives us: \( (x^{3}+6 x^{2})+(-2 x-12)\).

Step 2: Factor Out Common Elements

Now, factor out the common factor within each group. In the first group \( x^{3}+6 x^{2} \), \( x^{2}\) is a common factor. After factoring \(x^{2}\) out, our equation becomes \(x^{2}(x+6) -2(x+6)\).

Step 3: Factor Out Common Binomial

Finally, we factor out the common binomial, which is \( (x+6) \). Resulting in: \( (x^{2}-2)(x+6)\). This is the factored form of the original expression by grouping.

Key Concepts

Polynomial Factorization

Polynomial factorization is all about breaking down complex expressions into simpler multiplicative components. In the world of algebra, polynomials can often appear intimidating due to their complicated structures. However, by factoring them, we see that these algebraic expressions can be represented as products of simpler expressions, called factors. This process resembles breaking down numbers into their prime factors. The significant difference is that, with polynomials, we're working with variables. Factorization is not just a procedural task; it's a technique that opens the door to understanding the properties of polynomials better. This method is crucial for simplifying equations, simplifying arithmetic with algebraic expressions, and solving higher-degree polynomial equations.

Grouping Method

The grouping method is a strategic way of factoring polynomials that are not easily factorable using simpler methods. With polynomial expressions that contain more than three terms, direct factorization can be challenging. That's where grouping comes into play. By rearranging and grouping terms, we can identify simpler polynomial patterns hidden within a larger polynomial. This method works perfectly on expressions where no common factor exists for all terms, but instead, common factors are found in separate groups. Here's a simple breakdown of the grouping process:

• Identify natural groups within the expression (usually after rearranging terms).
• Factor out common terms within these groups.
• Look for common binomials across the groups to pull out.

Through this systematic approach, the grouping method converts complex expressions into manageable parts that reveal the overall structure of the polynomial.

Common Factors

Identifying common factors is a fundamental skill in algebra. These are elements that can multiply to form each term within a group or a polynomial. Spotting these shared elements is crucial in simplifying expressions and especially mandatory during factorization. Here's how to identify and work with common factors in polynomial expressions:

• Examine each term within a polynomial or group for like terms or constants.
• Factor these elements out to simplify each group or expression.

For instance, in the expression \(x^3 + 6x^2\), the common factor is \(x^2\), simplifying the term to \(x^2(x+6)\). Similarly, in expressions like \(-2x - 12\), the common factor is \(-2\), which transforms it into \(-2(x+6)\).Identifying these factors gives you a look inside the expression's hidden symmetry and allows easier manipulation into its fully factored form.

Binomials

Binomials are algebraic expressions containing two distinct terms joined by either addition or subtraction. They play a crucial role in polynomial factorization, especially in identifying common factors across grouped terms.A key example of binomials is terms like \((x+6)\), which appeared in the solution to the given exercise. The beauty of binomials lies in their simplicity and versatility; they can quickly become building blocks of larger expressions. When factoring polynomials, look for these bi-term structures after identifying common factors within individual groups. Here's a quick guideline to understand their role in grouping methods:

• After factoring out common factors from groups, observe if similar binomials emerge in different sections.
• Factor out the common binomial to simplify the polynomial further.

By recognizing and utilizing binomials, complex polynomial expressions are distilled into clear and manageable forms, aiding significantly in algebraic problem-solving.