Problem 33 - Elementary and Intermediate Algebra

Question

Find the GCF of each list of terms. $$ 24 a^{2}, 16 a^{3} b, 40 a b $$

Step-by-Step Solution

Answer

The GCF of the terms is \(8a\).

Step 1: Identify the Coefficients

The coefficients of the given terms are 24, 16, and 40. We need to find the greatest common factor (GCF) of these numbers.

Step 2: Find the GCF of the Coefficients

To find the GCF of 24, 16, and 40, list their factors: - Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 - Factors of 16: 1, 2, 4, 8, 16 - Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40 The common factors are 1, 2, 4, and 8. The greatest is 8, so the GCF of the coefficients is 8.

Step 3: Identify the Variables and Their Exponents

The variables present in the terms are \(a\) and \(b\). The exponents for \(a\) are 2, 3, and 1, respectively, and for \(b\), the exponents are 0 (not present), 1, and 1.

Step 4: Find the GCF of the Variables

For variable \(a\), take the lowest exponent: \(a^1\). For variable \(b\), take the lowest exponent: \(b^0\), but since \(b^0 = 1\), it doesn't affect the GCF as a factor.

Step 5: Combine the GCFs of Coefficients and Variables

Multiply the GCF of the coefficients by the GCF of the variables from the previous steps. The GCF of the coefficients is 8 and for \(a\) is \(a^1\). Thus, the combined GCF is \(8a\).

Key Concepts

Coefficients

In mathematics, a coefficient is the numerical factor present in an algebraic term. For example, in the term \(24a^2\), the number 24 is the coefficient. Coefficients are important as they help in multiplying a variable, assigning it a value by which it is scaled. Identifying coefficients correctly is crucial when working with polynomials and other algebraic expressions.

To find the Greatest Common Factor (GCF) of coefficients, you first need to list out the factors for each number. Factors are the integers that divide into the number without leaving a remainder. Here’s how you can do it:

• List all factors of each coefficient.
• Identify the common factors shared by these numbers.
• Select the greatest of these common factors.

This process is essential in simplifying expressions or when you are attempting to factor algebraic expressions in order to solve them.

Variables and Exponents

Variables are symbols like \(a\) and \(b\) that stand for numbers we don't know yet or numbers that can change. Exponents show how many times we need to multiply the variable by itself. In the expression \(16a^3b\), \(a\) is raised to the power of 3. This means \(a\times a\times a\). Understanding the roles of variables and exponents is key in simplifying algebraic expressions and in finding the GCF.

To find the GCF of variables across different terms, follow these steps:

• List out all the variables present in the expressions.
• For each variable, observe the different exponents in all terms.
• Select the smallest exponent across all terms for each variable as part of the GCF.

The reason you pick the smallest exponent is that it represents the maximum number of times that variable can divide into all the terms without leaving a remainder. In this example, the smallest exponent for \(a\) is 1, so the GCF would include \(a^1\). The "if-not-present" rule simplifies the process for variables that do not appear in every term, as they have an implied exponent of 0.

Factoring

Factoring is the process of breaking down an expression into products of other simpler expressions, which can multiply together to form the original expression. Knowing how to factor is crucial for simplifying algebraic expressions, solving equations, and understanding polynomials better.

When you factor something like \(24a^2, 16a^3b, 40ab\) to find the GCF, you’re essentially pulling out the largest possible expression that can divide each term evenly. This helps in reducing complexity and grouping expressions more efficiently.

Factoring involves:

• Identification of common numerical factors (coefficients) across terms.
• Recognizing the lowest power of variables common in all terms.
• Combining these elements to form the complete GCF.

Once you find the GCF, you can simplify or rewrite the original expression by dividing each term by this GCF, which often makes solving equations more manageable. Factoring is a core skill that repeatedly appears in algebra, whether you are dealing with polynomials or solving complex equations.