Problem 34 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 5 a b-8 a b c $$

Step-by-Step Solution

Answer

The completely factored expression is \( ab(5 - 8c) \).

Step 1: Factor by Identification

First, identify the common factor in both terms of the expression. In the terms \(5ab\) and \(-8abc\), the common factor is \(ab\).

Step 2: Factor Out the Common Term

Factor out \(ab\) from the expression.The expression becomes:\[ ab(5 - 8c) \]

Step 3: Simplification Check

After factoring, check if the expression inside the parentheses, \(5 - 8c\), can be further factored. Since \(5 - 8c\) cannot be factored further, it is left as it is.

Key Concepts

Common Factor

Understanding common factors is fundamental in algebra. A common factor is essentially a number or variable that is shared by all terms in an expression. Identifying it is the first crucial step in simplifying expressions through factoring. For example, in the expression \(5ab - 8abc\), both terms share the common factor \(ab\). This means that \(ab\) is a part of both \(5ab\) and \(-8abc\). Identifying these shared components helps in reducing expressions to their simpler forms. When you recognize a common factor, you can "factor it out," meaning you rewrite the expression to show this shared element outside of a parenthesis. This often simplifies the expression significantly, making further calculations or manipulation much easier.

Factoring Process

The factoring process involves systematically breaking down an algebraic expression into simpler parts that, when multiplied together, produce the original expression. It's akin to taking something complex and finding its basic building blocks, which can be quite helpful in solving equations. Here's a step-by-step guide to successful factoring:

• Identify and extract the common factor. For example, in the exercise \(5ab - 8abc\), the common factor is \(ab\), which is factored out to simplify the expression.
• Write down the common factor outside a set of parentheses. In our case, this results in \(ab(5 - 8c)\).
• Ensure that what's left inside the parentheses is in its simplest form. This may sometimes require additional factoring. However, in our case, \(5 - 8c\) does not need any further factoring.

By following these steps, you can successfully simplify complex algebraic expressions through factoring.

Simplification

Simplification in mathematics refers to the process of rewriting expressions in their most concise and easily interpretable form. This is often a final step in the factoring process. Once you've factored an expression, like \(5ab - 8abc\) into \(ab(5 - 8c)\), your next task is to check if further simplification is possible. You examine the expression inside the parentheses to see if it can be factored or reduced further. In this instance, \(5 - 8c\) is as simple as it gets.Simplification can often yield expressions that are easier to work with, allowing for more straightforward mathematical operations and analysis. Always remember, a fully simplified expression can make subsequent problems much easier to tackle.