Problem 17 - A Graphical Approach to College Algebra

Question

Factor each polynomial by grouping. $$t^{3}+2 t^{2}-3 t-6$$

Step-by-Step Solution

Answer

The factorized form is \((t^2 - 3)(t + 2)\).

Step 1: Group the Terms

First, we will split the polynomial into two groups. Group the first two terms and the last two terms:\[ (t^3 + 2t^2) + (-3t - 6) \]

Step 2: Factor Each Group

Next, factor out the greatest common factor from each group. In the first group, \(t^2\) is the greatest common factor, and in the second group, the greatest common factor is \(-3\):\[ t^2(t + 2) - 3(t + 2) \]

Step 3: Factor the Common Binomial

Notice that both terms now contain a \((t + 2)\), which we can factor out:\[ (t^2 - 3)(t + 2) \]

Step 4: Confirm the Factorization

Check that the factorization is correct by expanding the factors to see if we get back the original polynomial:\[(t^2 - 3)(t + 2) = t^3 + 2t^2 - 3t - 6 \] which matches the original expression.

Key Concepts

Grouping Method

The grouping method is a strategic approach for factoring polynomials, especially when dealing with four terms. It involves organizing terms into pairs or groups that can be factored more easily. In our exercise, we begin by grouping the polynomial \(t^3 + 2t^2 - 3t - 6\) into two parts: \((t^3 + 2t^2)\) and \((-3t - 6)\). This method prepares us to identify common factors within each group, setting the stage for further simplification.

• Step 1: Examine the polynomial and identify groupable terms.
• Step 2: Group the terms that will have a common factor.

By doing this, we create smaller expressions within the polynomial that are easier to manage. This simplification through grouping is particularly useful when direct factoring seems complex.

Greatest Common Factor (GCF)

The concept of the Greatest Common Factor (GCF) is important in simplifying expressions and equations. It refers to the largest factor shared by the numbers or terms in question. In the context of polynomial expressions, finding the GCF helps reduce complexity and prepare the polynomial for further factoring. For example, after grouping the polynomial terms \((t^3 + 2t^2) + (-3t - 6)\), we can identify:

• First Group: \(t^3 + 2t^2\), where the GCF is \(t^2\).
• Second Group: \(-3t - 6\), where the GCF is \(-3\).

Factoring these gives us \(t^2(t + 2) - 3(t + 2)\). Recognizing the GCF allows us to simplify expressions significantly, paving the way for the next step in factoring.

Binomial Factoring

Binomial factoring is a technique often used to simplify polynomials after initial grouping and GCF extraction. In this method, we look for common binomial factors within the expression. Once the terms are grouped and GCFs are extracted, like in the expression \(t^2(t + 2) - 3(t + 2)\), we often find similar or identical binomial factors across the groups.The final step is simple:

• Identify the Common Binomial: Look for a term shared by both expressions. Here, it is \((t + 2)\).
• Factor Out the Binomial: Remove it from each term to get the product \((t^2 - 3)(t + 2)\).

This process transforms the expression into a more manageable form, making it easier to solve or simplify further. The ability to spot and factor these common binomials is a powerful tool in polynomial algebra.