Problem 93 - Algebra for College Students

Question

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. $$2 x y+6 x+y+3$$

Step-by-Step Solution

Answer

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

Step 1: 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 \(y + 3\).

Step 2: 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.

Step 3: Identify common binomial factor

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

Step 4: 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.

Step 5: 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.

Key Concepts

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.