Problem 30 - Elementary Algebra

Question

Factor out the GCF. $$ 27 x-9 $$

Step-by-Step Solution

Answer

The factored expression is \(9(3x - 1)\).

Step 1: Identify the GCF

The greatest common factor (GCF) is the largest number that divides both terms. Here, the coefficients are 27 and 9. Both are divisible by 9.

Step 2: Write the GCF Outside

Write the GCF, which is 9, outside of a set of parentheses. This prepares us to factor out the GCF from both terms.

Step 3: Divide Each Term by the GCF

Divide each term of the expression by the GCF, 9. For the term \(27x\), divide: \(\frac{27x}{9} = 3x\). For \(-9\), divide: \(\frac{-9}{9} = -1\).

Step 4: Write the Factored Expression

The expression can now be written as the product of the GCF and the results from Step 3: \(9(3x - 1)\).

Key Concepts

Greatest Common Factor

The greatest common factor, or GCF, is a basic algebra concept that allows us to simplify expressions. It is the largest number that divides all terms of a given algebraic expression evenly. For instance, in the expression \(27x - 9\), the numbers 27 and 9 both have a GCF of 9.
To determine the GCF:

• List out all factors of each coefficient.
• Identify the largest factor that appears in each list.

Finding the GCF is a fundamental step in simplifying or factoring expressions and helps to make these expressions easier to work with in future calculations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the foundation of algebra and can be as simple as the exercise given, \(27x - 9\). Each part of an algebraic expression, such as a term or factor, plays a role in its manipulation and simplification.
These expressions are built from:

• Constants, like the 9 in our example.
• Variables, like \(x\), representing unknown values.
• Coefficients, such as the 27, that multiply the variables.

Understanding how these components work together is essential for performing operations like adding, subtracting, and especially factoring.

GCF Factoring

GCF factoring involves expressing an algebraic expression as a product of its GCF and another algebraic expression. This is done by factoring out the greatest common factor from each term, as seen in our solution example for \(27x - 9\).
The process includes:

• Identifying the GCF of all terms in the expression.
• Placing the GCF outside a set of parentheses.
• Dividing each term by the GCF to determine the terms inside the parentheses.

In our task, the GCF is 9, so factoring yields \(9(3x - 1)\). This simplified version is easier to manage in further calculations.

Basic Algebra Concepts

Basic algebra concepts provide the building blocks for more complex mathematical procedures. These include understanding components like terms, coefficients, and constants, and performing operations such as addition, subtraction, multiplication, and division.
Some foundational skills derived from basic algebra are:

• Identifying and working with variables and coefficients.
• Understanding and applying operations such as equation solving and factoring.
• Simplifying expressions by combining like terms or using the GCF.

These skills facilitate not just solving specific problems but also building a groundwork for tackling high-level algebra and other areas of mathematics.