Problem 45 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises, factor the polynomial. If the polynomial is prime, state it. $$ 3 a x+6 a y+b x+2 b y $$

Step-by-Step Solution

Answer

The factored form of the given polynomial \(3ax + 6ay + bx + 2by\) is \((3a + b)(x + 2y)\).

Step 1: Group the terms

To begin, we will need to group the terms of the polynomial. To make it easier to identify the common factors, we can group the terms as follows: $$ (3ax + 6ay) + (bx + 2by) $$

Step 2: Identify common factors in each group

Next, we need to identify the common factors within each group that we can factor out. In the first group, we can see that the common factor is \(3a\), while the common factor in the second group is \(b\).

Step 3: Factor out the common factors

Now that we have identified the common factors within each group, we can factor them out by dividing each group by the common factor. This will give us: $$ 3a(x + 2y) + b(x + 2y) $$

Step 4: Factor out the common term

We can now see that the expression \((x + 2y)\) is common to both terms. We will factor out this expression, and we have: $$ (3a + b)(x + 2y) $$ Now the polynomial is factored into its simplest form.

Key Concepts

Grouping Method

The grouping method is a strategy used in algebra to simplify and factor polynomials when there are four or more terms involved. This method works by rearranging and grouping terms so that we can find common factors within each group. To apply this technique, we start by observing the given polynomial and looking for ways to split it into smaller groups where common factors are more evident.

Let's take the example problem: \(3ax + 6ay + bx + 2by\). The first step is to group similar terms, which can be done in various ways depending on the particular expression. In our case, we group the terms with similar coefficients or variables. Once the terms are grouped—like \(3ax + 6ay\) and \(bx + 2by\)—we can then proceed to look for common factors in each of these subsets. The successful application of the grouping method makes factoring more manageable by breaking down a complex polynomial into simpler components.

Common Factors

Identifying common factors is an essential step in factoring polynomials. A ‘factor’ in algebra is anything multiplied to form a product. A common factor is an expression that divides evenly into all terms of the polynomial we're trying to factor. In essence, to find the common factors, we look for numbers, variables, or combinations of both that appear in every term of the polynomial.

From our polynomial \(3ax + 6ay + bx + 2by\), we see that in the group \(3ax + 6ay\), the common factor is \(3a\), and in \(bx + 2by\), the common factor is \(b\). Extracting these out simplifies the expression inside each group, leading to \(3a(x + 2y) + b(x + 2y)\). By recognizing these common factors, we effectively set the stage for the final factoring, where the common factors play a pivotal role in simplifying the algebraic expression to its most reduced form.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. Polynomials are a specific type of algebraic expression that consist of terms added or subtracted together, each composed of variables raised to whole-number exponents and their coefficients.

When factoring polynomials like \(3ax + 6ay + bx + 2by\), we treat each term as a part of a larger puzzle. By breaking down the expression into its constituent parts and isolating the common factors, we eventually reassemble the expression into a product of simpler expressions.

Understanding Algebraic Structure

In our case, we arrived at the factored form \(3a(x + 2y) + b(x + 2y)\), then noticed the common binomial \(x + 2y\) to obtain the final factored polynomial \(3a + b)(x + 2y)\). By understanding the structure and components of algebraic expressions, we gain the ability to manipulate and simplify these expressions significantly, a fundamental skill in algebra.