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Chapter 3: Problem 93

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$2 x y+6 x+y+3$$

Short Answer

Expert verified
The polynomial factors as \((y+3)(2x+1)\).

Step by step solution

01

Identify like terms

First, we will rearrange the terms to identify any possible groupings. The expression given is \(2xy + 6x + y + 3\). Let's notice the pair \(2xy + 6x\) and \(y + 3\).
02

Factor out the common factor in each group

From the first group \(2xy + 6x\), we can factor out a \(2x\), giving us \(2x(y + 3)\). In the second group \(y + 3\), there is no need to factor further as it is already simple.
03

Identify common binomial factor

Both grouped terms, \(2x(y+3)\) and \(1(y+3)\), have \((y+3)\) as a common binomial factor.
04

Factor out the common binomial

We factor \((y+3)\) out of the entire expression: \[ (y+3)(2x+1) \]. This is the factored form of the polynomial.
05

Confirm the factorization

Distribute to check: Expanding \((y+3)(2x+1)\) gives:\(y(2x+1) + 3(2x+1) = 2xy + y + 6x + 3\). This matches our original polynomial, confirming that the factorization is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Grouping Method
When dealing with polynomials, the grouping method is a valuable tool for factoring expressions when they do not readily fit simpler methods like factoring by a greatest common factor or using special formulas. Here's how it works:
  • Identify pairs: Start by organizing the terms of the polynomial into pairs or groups that seem to have something in common. In our example, the expression given is \(2xy + 6x + y + 3\). By inspecting the terms, we can consider grouping them as \((2xy + 6x)\) and \((y + 3)\).
  • Factor within groups: Once grouped, look within each group to see if you can factor out a common factor. For the first group \((2xy + 6x)\), the common factor is \(2x\). This gives us \(2x(y + 3)\).
  • Repeat as needed: If the other group \((y + 3)\) is already in its simplest form or shares no common factor, you can move forward.
The goal of the grouping method is to simplify the polynomial into parts that reveal a common factor, helping further factorization steps.
Common Factor
Finding a common factor is one of the most fundamental skills in algebra, especially in simplifying or factoring polynomials. The common factor is a number or variable that is shared among terms in an expression. Let's break it down further:
  • Understand the terms: Observe each term in the polynomial and determine any common numerical or variable components. For instance, in the group \(2xy + 6x\), both terms have \(2x\) as a common factor.
  • Extract the factor: Carefully pull out this common factor from the group, simplifying it to \(2x(y + 3)\) as shown in our example.
Performing these steps makes the problem cleaner and often unlocks easier paths to completely factor or simplify the polynomial. It's a building block process that leads to further simplification.
Binomial Factor
A binomial factor is a factor that consists of two terms. In factored polynomials, identifying a common binomial factor across terms can be a key step in simplifying the expression. Let's see how it applies:
  • Identify repeated elements: After initially simplifying each group, observe if there are repeated expressions in each factored form. For our polynomial, both groups result in the formula \((y+3)\).
  • Factor using the binomial: Since \((y+3)\) is common in both \(2x(y + 3)\) and \(1(y + 3)\), we extract this common binomial factor. This gives us the final factorized form: \((y+3)(2x+1)\).
  • Final check: Always verify by expanding back the factors to ensure they reconstruct the original polynomial correctly.
Recognizing a binomial factor simplifies polynomials significantly and aids in confirming the accuracy of your solutions.

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