Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Factor out the greatest common factor. $$ (3 a+b)(2 c-d)+2 a(2 c-d)^{2} $$

Step-by-Step Solution

Answer

The short answer is: \((2c-d)(3a+b+2a(2c-d))\).

Step 1: Identify the common factor in the terms of the expression

In our given expression, let's first look at each term and find the common factor. \( (3a+b)(2c-d)+2a(2c-d)^{2} \) Look at the two terms: 1. \((3a+b)(2c-d)\) 2. \(2a(2c-d)^{2}\) We can see that the expression \((2c-d)\) is present in both terms.

Step 2: Factor out the common factor

Now that we've identified the common factor \((2c-d)\), we'll rewrite the expression by factoring it out: \[ (2c-d)((3a+b) + 2a(2c-d)) \]

Step 3: Simplify the remaining expression

We now have the factored expression: \[ (2c-d)((3a+b) + 2a(2c-d)) \] or in other words \[ (2c-d)(3a+b+2a(2c-d)) \] The given expression has been factored with the greatest common factor.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \( x \), \( y \), or \( a \)), and operators (such as addition, subtraction, multiplication, and division). The purpose of algebraic expressions is to represent real-world situations algebraically so that we can solve various kinds of problems.

It's similar to a recipe that uses a combination of ingredients to make something new. In algebra, the ingredients are numbers and variables, and the recipe is the expression itself, telling us how to combine those ingredients. Whenever we work with algebraic expressions, we may be asked to perform operations like simplification, evaluation, or, as in our original exercise, factoring.

Greatest Common Factor

The greatest common factor (GCF), also known as the greatest common divisor, is the highest number that divides exactly into two or more numbers without leaving a remainder. When dealing with algebraic expressions, the GCF is not just restricted to numbers but can also include variables and even expressions.

Identifying the GCF in algebraic terms is like finding a common thread in a story. For the expression \( (3a+b)(2c-d)+2a(2c-d)^2 \), we noticed the common thread, or factor, was \( (2c-d) \). We can think of factoring out the GCF as the 'undoing' of distribution, gathering all the shared parts outside while revealing what's unique inside each term of the expression.

Polynomial Factoring

Polynomial factoring is the process of expressing a polynomial as the product of its factors, which are polynomials of lower degrees. This process often simplifies the polynomial and is particularly useful in solving equations, simplifying expressions, and finding polynomial roots.

Think of a polynomial like a multi-layered cake. Factoring breaks down the cake into its original layers, making it easier to see each individual part. In our exercise, we started with a polynomial expression that appeared quite complex. However, by factoring out the GCF, we separated the polynomial into two clear factors, making the expression easier to understand and work with in subsequent math operations.

Example Improvement

When learning polynomial factoring, remember that not every polynomial is factorable, and sometimes you're already looking at the simplest form. Also, for expressions containing multiple terms, always first look for a GCF to simplify the factoring process. The end goal is to write the expression as a product of its factors in the simplest possible terms.