Problem 55 - Elementary and Intermediate Algebra

Question

Factor. See Example 5 or Example 7. $$ 2 u^{2}-6 v^{2}-u v $$

Step-by-Step Solution

Answer

The expression cannot be factored further using basic polynomial methods.

Step 1: Identify the terms

The expression given is \(2u^2 - 6v^2 - uv\). It consists of three terms: \(2u^2\), \(-uv\), and \(-6v^2\). Our task is to factor this trinomial, if possible.

Step 2: Find common factors in pairs

Separate terms into groups that might share common factors. Start by looking at \(2u^2 - uv\) and \(-6v^2\) separately.For \(2u^2 - uv\), both terms have \(u\) as a common factor.Factor \(u\) out to get: \(u(2u - v)\).For \(-6v^2\), there is nothing to factor with respect to \(2u^2 - uv\), so this term remains as it is.

Step 3: Identify common factors

Check if the resulting expression \(u(2u - v) - 6v^2\) can be expressed in terms of a common binomial factor. Notice however, that we are unable to group this term into a form that would allow a straightforward factoring because they do not result in a common binomial factor, therefore no further factoring can be performed.

Step 4: Verify incompleteness

Since grouping did not result in an easily factorable form and there are no apparent common factors across all terms other than 1, the expression \(2u^2 - 6v^2 - uv\) cannot be factored using simple polynomial factoring techniques. We suggest double-checking the arrangement of terms and any potential errors for compatibility in factorization - none exist here.

Key Concepts

Understanding Trinomials

A trinomial is a type of polynomial characterized by having three distinct terms. In the expression \(2u^2 - 6v^2 - uv\), we observe three parts: \(2u^2\), \(-6v^2\), and \(-uv\). Trinomials can vary significantly, and their structure determines the different strategies we might employ to factor them.

• The structure of a trinomial can suggest possible methods for factoring. For example, is there a pattern similar to the difference of squares or a perfect square trinomial?
• Understanding the arrangement and the consistency of terms is crucial. Sometimes rearranging can help in better visualizing common elements or applying different factoring techniques.
• When approaching trinomials, recognizing whether it's quadratic or if it belongs to another polynomial family can guide the selection of correct factoring methods.

The key takeaway is to comprehend the unique nature of the trinomial, as this understanding is foundational for the subsequent steps in factoring.

Role of Common Factors

Common factors are elements that divide all the terms in a polynomial without leaving a remainder. In the trinomial \(2u^2 - 6v^2 - uv\), initially, we look for any factors all terms might share.

• The most basic common factor is always 1, which is the only universal factor when no other number divides all terms.
• Here, you can identify that terms like \(2u^2\) and \(-uv\) have \(u\) in common, which can be factored out from their group.

Finding common factors is a pivotal step. It simplifies parts of the polynomial and makes it easier to identify other factoring techniques. If no significant common factors exist across all terms, we reassess using grouping or other methods.

Effectiveness of the Grouping Method

The grouping method is a handy technique when a polynomial doesn't present immediate straightforward factors. It involves rearranging terms to expose common factors within grouped sections. For \(2u^2 - 6v^2 - uv\):

• By examining sub-groups like \(2u^2 - uv\), we extract common factors, simplifying to \(u(2u-v)\).
• However, the standalone term \(-6v^2\) doesn't fit neatly into a group with the same factor already found.

Attempting the grouping method reveals limitations in the presence of terms that resist pairing through common factors or when no rearrangement yields a common binomial factor. Expertise takes time and practice, and not all expressions are factorable by simple methods. This method shines when you're successful in reshuffling terms to unearth hidden commonalities.