Problem 40 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. $$a(9 c+4)-b(9 c+4)$$

Step-by-Step Solution

Answer

The short answer is: \((9c+4)(a - b)\)

Step 1: Identify the greatest common factor

The given expression is \(a(9c+4) - b(9c+4)\). Observe that both terms have the common factor \((9c+4)\).

Step 2: Factor out the common factor

Now that we have identified the common factor \((9c+4)\), we can factor it out from the expression. This is done by rewriting the expression as a product of the common factor and another expression: \((9c+4)(a - b)\)

Step 3: Check the answer

To make sure our solution is correct, we can expand the factored expression to see if it matches the original one: \((9c+4)(a-b) = (9c+4)a - (9c+4)b = a(9c+4) - b(9c+4)\) Since the factored form and the original expression match, we can be confident that our solution is correct. So, the factored expression is: \((9c+4)(a - b)\)

Key Concepts

Understanding the Greatest Common Factor (GCF)

When faced with expressions or numbers, one useful technique is to find the Greatest Common Factor (GCF). The GCF is the largest factor that two or more numbers or terms have in common. For expressions, it involves identifying the common divisors of the terms within the expression.

Imagine you have two numbers, like 8 and 12. Their factors are:

• 8: 1, 2, 4, 8
• 12: 1, 2, 3, 4, 6, 12

The common factors are 1, 2, and 4. Among these, the greatest is 4, making it the GCF.

For algebraic expressions, finding the GCF is similar but involves variables and coefficients. In the exercise expression \(a(9c+4) - b(9c+4)\), \((9c+4)\) is the common factor for both terms. By factoring it out, calculations and simplifications become easier.

Algebra and Expression Manipulation

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It's the language of mathematics that allows us to express general mathematical ideas on how different objects relate.

Algebraic expressions are key components, composed of numbers, variables (letters), and operations (like addition and subtraction). In the given expression \(a(9c+4) - b(9c+4)\), we deal with two terms involving multiplication and subtraction.

The ability to manipulate and simplify these expressions is crucial. For example, recognizing common factors can lead to simplified expressions, which are easier to evaluate and solve. In fact, utilizing the concept of factoring, we efficiently reformulate our expression into \((9c+4)(a - b)\) by recognizing the shared part of both terms.

Mastering Expanding Expressions

Expanding expressions is essentially the opposite of factoring them. It involves taking a factored expression and distributing the factors across the terms to reach the original state. This step is key when checking your work in algebra.

In our problem, once the expression is factored as \((9c+4)(a-b)\), we expand it to verify correctness by distributing:

• \((9c+4) \times a\) becomes \(a(9c+4)\)
• \((9c+4) \times -b\) becomes \(-b(9c+4)\)

Adding these back together, we get the original expression \(a(9c+4) - b(9c+4)\). Expanding not only verifies the factorization but also enhances comprehension of the algebraic structure. By mastering expanding, you gain confidence that your factorization has been executed correctly.