Problem 30 - Elementary and Intermediate Algebra

Question

Find the GCF of each list of terms. $$ 18 r, 27 r^{3} $$

Step-by-Step Solution

Answer

The GCF of the terms is 9r.

Step 1: Identify the coefficients

Look at the numerical part of each term. For the terms 18r and 27r^3, the coefficients are 18 and 27.

Step 2: Find the GCF of coefficients

Identify the greatest common factor (GCF) of the coefficients 18 and 27. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 27 are 1, 3, 9, and 27. The largest common factor is 9.

Step 3: Identify the variables and their powers

Look at the variable part of each term. The variables are both r, but with different powers: r^1 in 18r and r^3 in 27r^3.

Step 4: Determine the smallest power of the variable

The smallest power of the variable r that is common in both terms is r^1.

Step 5: Combine the results

Multiply the GCF of the coefficients (9) by the smallest power of the variable, r^1, to get the overall GCF. Thus, the GCF of 18r and 27r^3 is 9r.

Key Concepts

Coefficients in Algebra

When dealing with algebraic terms, it's important to understand what coefficients are. Coefficients are numerical values that are multiplied by variables in an algebraic expression. Take, for example, the terms in our original exercise: \(18r\) and \(27r^3\). Here, the numbers 18 and 27 are the coefficients. These numbers essentially determine the size of the term in relation to the variable part. In algebra, coefficients can be any real number: positive, negative, or even a fraction. Identifying and understanding coefficients is crucial because they are central to operations like addition, subtraction, and especially factoring—processes that require clear recognition of each term’s size. Why focus on coefficients?

• They help simplify complex expressions by focusing on the numerical part.
• They are key to determining the Greatest Common Factor (GCF) of terms.
• Understanding them helps in combining like terms effectively.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Unlike an equation, an expression doesn't have an equal sign and can't be solved but can be simplified or factored. Our original exercise involves the algebraic expressions \(18r\) and \(27r^3\), both consisting of a coefficient and a variable part. Components of algebraic expressions:

• Coefficients: As discussed earlier, these are the numerical factors.
• Variables: Symbols that represent unknown values, often denoted as letters like \( r \) in our example.
• Constants: Fixed numbers without any variables, though not present in this particular exercise.

Algebraic expressions can represent real-world situations and are crucial for understanding and solving problems in calculus, physics, and economics. Learning how to handle them is foundational to mastering algebra.

Factoring in Algebra

Factoring is the process of breaking down an expression into a product of simpler expressions. It's a fundamental technique used to simplify algebraic expressions and solve equations. For the terms \(18r\) and \(27r^3\), we focused on finding the Greatest Common Factor (GCF), which is an essential step in factoring.

Steps in Factoring:

• Identify the GCF: The largest factor that divides all terms in an expression. In our exercise, the GCF of the coefficients 18 and 27 is 9.
• Examine the variables: Look at the powers of the variables in each term. Use the smallest power common to both terms, like \( r^1 \) in this case.
• Multiply GCF with the smallest variable power: This gives the complete GCF, \( 9r \) for our terms.

Factoring helps in simplifying expressions and is also used to solve quadratic equations, work with polynomials, and even in partial fraction decomposition. Mastering factoring is immensely beneficial for progressing in algebra and higher mathematical concepts.