Problem 47 - Algebra for College Students

Question

Factor by grouping. $$a x+4 x+a y+4 y$$

Step-by-Step Solution

Answer

The factored form is \((a + 4)(x + y)\).

Step 1: Group the Terms

First, let's arrange the expression into groups. We'll group the terms as follows: \((ax + 4x)\) and \((ay + 4y)\). This will help us factor by grouping in the next step.

Step 2: Factor Out the Common Factor in Each Group

Look at the first group \(ax + 4x\). The common factor here is \(x\). Factoring \(x\) out, we have: \(x(a + 4)\). Next, examine the second group \(ay + 4y\). The common factor here is \(y\). Factoring \(y\) out, we have: \(y(a + 4)\).

Step 3: Combine the Factored Groups

Now, observe that both terms, \(x(a + 4)\) and \(y(a + 4)\), have a common binomial factor \((a + 4)\). So we can factor \((a + 4)\) out from the expression, yielding: \((a + 4)(x + y)\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a valuable skill in algebra that simplifies expressions and solves equations more easily. Essentially, it involves expressing the polynomial as a product of simpler factors. The core idea is to identify patterns or common elements within the polynomial that can be grouped and factored out. This is often the first step before breaking down the expression further or solving it.

Polynomials can have different numbers of terms, and factoring approaches might differ based on this. For example, the polynomial from the exercise is split into two binomial groups, making it easier to identify and extract common factors. The outcome simplifies the expression, revealing hidden relationships between its terms. Remember, factoring requires practice, but it's a skill that sharpens your problem-solving abilities in algebra.

Common Factors

The concept of common factors is central to simplifying mathematical expressions, especially in the context of factoring polynomials. A common factor is a number or variable that divides two or more terms without a remainder. Recognizing these can make solving complex expressions easier as they can be factored out, reducing the complexity of the expression.

In the exercise provided, each grouping

• The group \(ax + 4x\) has \(x\) as a common factor.
• The group \(ay + 4y\) has \(y\) as a common factor.

Once these common factors are factored out, the expressions become \(x(a + 4)\) and \(y(a + 4)\). Understanding common factors is essential because it leads to recognizing further opportunities for simplification, often revealing shared factors between groups, as seen with the \(a + 4\) term here.

Binomial Expressions

Binomial expressions are algebraic expressions with exactly two terms, connected by a plus or minus sign. These are quite common in algebra, and understanding how to manipulate them can significantly enhance your ability to work through factoring problems and simplify expressions.

In the original problem, after grouping and factoring out common terms, the structure of binomial expressions becomes apparent, specifically \((a + 4)\) as a common binomial factor for both groups. Binomials are particularly significant in factoring because they often serve as factors that simplify polynomial expressions.

• When both groups in the problem have a common binomial expression, it's factored out, leading to the final simplified expression.

Recognizing and working with binomial expressions is a key aspect of algebra, providing a pathway to simplifying and solving polynomial equations efficiently.