Problem 46 - Elementary and Intermediate Algebra

Question

Factor by first grouping the appropriate terms. \(s^{2}-t^{2}+s-t\)

Step-by-Step Solution

Answer

The expression factors to \(s(s+1) - t(t+1)\).

Step 1: Rearrange the Expression

Start by rearranging the terms to: \[ (s^2 + s) - (t^2 + t) \]This rearrangement will help us see the common factors more easily when we group them.

Step 2: Group the Terms

Group terms in pairs:\[ (s^2 + s) - (t^2 + t) \]This expression groups terms based on similar structures like \[ (s(s + 1)) - (t(t + 1)) \].

Step 3: Factor Each Group

Now, factor out the common factors from each group:\[ s(s+1) - t(t+1) \]In each pair, the common factor is the variable term.

Step 4: Use the Difference of Products

Notice that we now have a difference of two products:\[ s(s+1) - t(t+1) \]Keep the expression as it is since it is factored as a difference of products.

Key Concepts

Grouping Terms

Grouping terms is a pivotal technique in factoring algebraic expressions, especially when dealing with polynomials. This method involves organizing terms into pairs or groups, so they exhibit a common structure. It simplifies the process of identifying and extracting common factors from these groups.
Grouping is particularly useful when you cannot immediately spot a common factor across all elements of the expression. Instead, by separating terms into strategic groupings, you can find factors within each group, which might otherwise not be obvious.

• For example, in the given expression, \(s^2 - t^2 + s - t\), grouping similar structures can streamline the factoring process.
• By rearranging and grouping, you reorganize terms to allow for more straightforward identification of factors, leading to an easier simplification process.

When applying grouping, focus on logical connections between terms, such as shared variables or coefficients, to decide the optimal way to create meaningful groups.

Rearranging Expressions

Rearranging expressions is a fundamental skill in algebra that prepares an equation or expression for further manipulation, like factoring. Through rearrangement, you can highlight similarities between different parts of the expression, making it easier to apply various algebraic techniques.
Rearrangement typically involves changing the order of terms in a polynomial. The goal is to place terms with similar characteristics next to each other to readily spot patterns or common elements. For instance, rearranging the expression \(s^2 - t^2 + s - t\) as \((s^2 + s) - (t^2 + t)\) arranges the terms to help identify potential groupings based on their structures.

• This organization is crucial as it reveals symmetries or similar patterns—steps that pave the way for efficient grouping and factoring.
• Beyond revealing common factors, rearranging can sometimes simplify the cognitive load by aligning terms in a visually manageable manner.

Thus, rearrangement is a strategic maneuver in algebra, setting the groundwork for subsequent operations like grouping and factoring.

Common Factors

Identifying common factors in polynomial expressions is key to simplifying or factoring them. A common factor is any variable or coefficient that appears in all terms within a group you are examining.
Finding common factors involves:

• Recognizing repeated numbers or variables across terms.
• Extracting these repeated elements to simplify the expression or prepare it for further factoring techniques.

In the case of \((s^2 + s) - (t^2 + t)\), identifying \(s\) and \(t\) as common across their respective groups means that each can be factored out, resulting in \(s(s+1) - t(t+1)\).
Mastering the art of spotting and utilizing common factors paves the way to more advanced calculations and equation solving. It streamlines a seemingly complex expression into more manageable parts, significantly aiding in its simplification.