Problem 7 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Factor out the greatest common factor. $$ 3 x(2 x+1)-5(2 x+1) $$

Step-by-Step Solution

Answer

The greatest common factor (GCF) of the given expression is (2x + 1). By factoring out the GCF, we rewrite the expression as the product \( (2x + 1)(3x - 5) \).

Step 1: Identify the GCF of the terms

We first look at the given expression and identify the greatest common factor (GCF) of the terms within the expression. In this case, it is (2x + 1), which can be found in both terms: $$3x(2x+1)$$ and $$-5(2x+1)$$.

Step 2: Rewrite the expression as a product of the GCF

With the GCF identified as (2x + 1), we can rewrite the expression as the product of the GCF and the sum of the remaining terms from the original expression. This will involve using the distributive property in reverse (factoring). The expression can be rewritten as: $$ (2x + 1)(3x - 5) $$

Step 3: Simplify

Since there are no more common factors between the remaining terms in the product, we have simplified the original expression by factoring out the greatest common factor, resulting in the final factored form: $$ \boxed{(2x + 1)(3x - 5)} $$

Key Concepts

Greatest Common Factor (GCF)

To understand factoring, it's important to get a grip on what the Greatest Common Factor (GCF) means. The GCF is simply the largest factor that two or more terms have in common.
In our exercise, we're working with terms that can be grouped together by their common factors. This step helps in reducing algebraic expressions into simpler parts.
In our example, the expression is \( 3x(2x+1) - 5(2x+1) \). The GCF here is \( (2x + 1) \), as it appears in both terms. When you "factor out" the GCF, you're pulling out this common element from each term, simplifying the entire expression.
Knowing how to identify the GCF is crucial because it is often the first step in both factoring and simplifying complex algebraic expressions.

• Locate similar or identical parts in different terms.
• Factor out this common element to simplify the expression.

Recognizing and factoring out the GCF makes expressions manageable and serves as a foundation for more advanced algebra.

Distributive Property

The distributive property is a fundamental rule in algebra. It involves distributing multiplication over addition or subtraction within parentheses.
In essence, it allows you to multiply a term across terms within parentheses effectively. For instance, in expressions like \( a(b + c) \), you expand it to \( ab + ac \).
In reverse, this means you can also factor out a common factor from terms inside parentheses. In our problem, the distributive property is used in reverse when we factor out the GCF \((2x + 1)\). By doing this, we turn \( 3x(2x+1) - 5(2x+1) \) into \((2x + 1)(3x - 5)\).
Here's how the distributive property plays out:

• Multiply each term inside the parentheses by the term outside, or factor one out.
• In this exercise, recognize terms that share a factor and "withdraw" it outside the parentheses.

In practice, the distributive property allows us to reorganize expressions and understand relationships between different components.

Algebraic Manipulation

Algebraic manipulation is the process by which we simplify or rearrange expressions to make them more understandable or solve them efficiently. It’s a skill that requires practice and a good grasp of fundamental algebraic principles.
In our example, algebraic manipulation was used to transform the expression \(3x(2x+1) - 5(2x+1) \) into its factored form \((2x+1)(3x-5) \). This was achieved by first identifying and factoring out the GCF and then using the distributive property correctly.
For effective algebraic manipulation:

• Understand the components of your expression.
• Use mathematical properties like commutative, associative, and distributive wisely.
• Break down the expressions and approach each part systematically.

Mastering algebraic manipulation can greatly aid in solving equations, simplifying expressions, and making complex problems more approachable.