Problem 76 - Elementary and Intermediate Algebra

Question

Factor. $$ a b(c-7)-12(c-7) $$

Step-by-Step Solution

Answer

The factored expression is \((c-7)(ab - 12)\).

Step 1: Identify the Common Factor

In the expression \( ab(c-7) - 12(c-7) \), the term \((c-7)\) appears in both parts of the expression. This is the common factor that can be factored out.

Step 2: Factor Out the Common Factor

Factor \((c-7)\) out of each term in the expression. This means we rewrite the expression as \((c-7)(ab - 12)\).

Key Concepts

Understanding Common Factors

A common factor is like a "shared ingredient" in mathematical expressions or terms. Imagine you are baking cookies, and some ingredients like flour or sugar are present in each dough mixture. Similarly, in algebraic expressions, certain terms appear in every part of the expression. These are your common factors. In our exercise, the expression was \( ab(c-7) - 12(c-7) \). Here, \((c-7)\) appears in both terms, making it a common factor. Recognizing these common ingredients helps simplify complex expressions.
To spot common factors:

• Look for repeated expressions or terms within the equation or expression.
• Check if there is a number or variable that consistently appears in different parts of the expression.
• Write down these repeated elements to see if they can be isolated through factoring out.

This approach simplifies solving and manipulating algebraic equations, making it much easier to work with and understand.

Algebra Problem Solving Essentials

Solving algebra problems might seem challenging, but with structured techniques, they become a lot more manageable. One crucial skill in algebra involves recognizing patterns and identifying opportunities to simplify through factoring. In the given problem, we use the common factor concept to factorize efficiently.
Steps to solve algebra problems:

• Identify all parts of the expression or equation. Break them down to understand their roles.
• Look for ways to simplify. This often involves finding common factors or using known algebraic identities.
• Apply mathematical operations carefully. This includes adding, subtracting, multiplying, or dividing terms consistently.

Understanding these fundamental steps provides a pathway for approaching any algebra problem, irrespective of its complexity. As you practice these steps, problem-solving will become less daunting.

Benefits of a Step-by-Step Solution

Breaking down problems into step-by-step solutions is like solving a puzzle one piece at a time. This systematic approach is especially helpful in algebra, where problems can appear overwhelming at first glance. The given exercise is a good example:
1. **Identify the common factor:** Recognized \((c-7)\) as a shared element, which is a crucial starting point.2. **Factor out the common terms:** This revealed a simplified expression, \((c-7)(ab - 12)\).
Why step-by-step solutions help:

• They make complex problems less intimidating by breaking them into smaller, manageable parts.
• Students can track their understanding and ensure each phase of the problem is handled correctly.
• It teaches meticulousness and patience, spreading confidence in problem-solving.

With practice, you will find that each small step leads to significant strides in understanding and mastery of algebra.