Problem 1 - College Algebra

Question

In Exercises \(1-10\), factor out the greatest common factor. $$18 x+27$$

Step-by-Step Solution

Answer

The factorised form of the expression \(18x + 27\) is \(9(2x + 3)\)

Step 1 Find the Greatest Common Factor (GCF)

Firstly, determine the GCF of the numerical coefficients. The numerical coefficients of the terms are 18 and 27. By observation, the GCF of 18 and 27 is 9.

Step 2 Factorise the Expression

The expression is now factorised by taking out the common factor, which is 9. So, divide each term of the expression by the common factor 9. The factorised expression becomes \(9(2x + 3)\)

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a key concept in mathematics that helps simplify algebraic expressions. It refers to the largest number that can evenly divide each term within the expression without leaving a remainder. Finding the GCF is essential when it comes to factoring polynomials, as it allows us to break down expressions into simpler components.
To determine the GCF of numerical coefficients, follow these steps:

• Identify all factors of each term's coefficient. For example, for 18, the factors are 1, 2, 3, 6, 9, and 18.
• List the factors for the other coefficient, such as 27, which are 1, 3, 9, 27.
• Compare the factors from both lists and choose the largest one that appears in both. For 18 and 27, this would be 9.

Understanding the GCF is the first step in many algebra problems and is foundational for simplifying complex expressions.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operators. They form the basis of algebra and can represent a variety of real-world problems. Learning how to manipulate and simplify these expressions is a fundamental skill in mathematics.
In an algebraic expression like \(18x + 27\), each part plays a role:

• The coefficients (such as 18 and 27) are the numerical parts that multiply the variables.
• The variable \(x\) represents an unknown value that can change.

A goal in algebra is often to simplify or 'factor' these expressions, making them easier to work with or solve. Factoring involves rewriting the expression as a product of its simplest forms. By factoring out the GCF, like in our example, we rewrite the expression in terms of simpler components \(9(2x + 3)\). This can make solving equations or further algebraic manipulations more straightforward.

Mathematical Problem Solving

Mathematical problem solving is more than just applying formulas; it's about understanding concepts and the relationships between different elements in a problem. Factoring, as an example, requires a strategic approach to identify common factors and simplify expressions.
When you encounter a problem like factoring \(18x + 27\), follow these steps to solve it efficiently:

• Identify the Problem: Recognize that you need to factor the expression.
• Gather information: Find the GCF as the first step since it will simplify the expression.
• Perform Operations: Divide each term by the GCF, leading to the factored form \(9(2x + 3)\).

By practicing these steps and understanding the "why" behind each action, you improve your problem-solving skills and become more confident in tackling different mathematical challenges. Always remember, problem solving in mathematics is greatly enhanced by first breaking down the problem into digestible parts before proceeding with the solution.