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Chapter 6: Problem 7

Factor the given expressions completely. $$5 a-5$$

Short Answer

Expert verified
The expression factors to \(5(a - 1)\).

Step by step solution

01

Identify Common Factors

First, look at the expression \(5a - 5\). Both terms, \(5a\) and \(-5\), have a common factor of 5. This means we can factor out the 5 from each term.
02

Factor Out the Common Factor

Remove the common factor of 5 from the expression. This gives: \[ 5(a - 1) \] This is because when you divide \(5a\) by 5, you get \(a\), and \(-5\) divided by 5 gives \(-1\).
03

Verify Your Factorization

Distribute the 5 back to check your work: \(5(a - 1) = 5a - 5\). The expression matches the original expression confirming that the factorization is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factors
In algebra, recognizing common factors is a key step in simplifying expressions. When exploring the expression \(5a - 5\), note that both terms, \(5a\) and \(-5\), share a common factor of 5. This factor is the largest number that evenly divides each term in the expression. By identifying common factors, you simplify the process of factoring and reduce complexity in equations.
  • What are Common Factors? - These are numbers or variables that are shared by all terms in an expression.
  • Finding Common Factors: - Examine each term to find the largest number or variable that divides each one without a remainder. For example, in \(5a\) and \(-5\), 5 is the greatest common factor.
Mastering common factors makes comprehending algebraic expressions easier. It's like looking for the most obvious piece that fits in all parts of a puzzle to simplify the picture. Remember, recognizing common factors can set the stage for solving more complex problems.
Step-by-Step Solutions
A step-by-step approach breaks down the process of factorization into understandable pieces. This methodical breakdown makes learning more digestible.
  • Step 1: Identify Common Factors - First, look for common factors, as explained earlier. In \(5a - 5\), the common factor is 5.
  • Step 2: Factor Out the Common Factor - Remove the common factor by dividing each term by it. So, dividing \(5a\) by 5 becomes \(a\), and \(-5\) divided by 5 becomes \(-1\). The expression simplifies to \(5(a - 1)\).
This detailed exploration helps highlight why each step is meaningful, offering a window into the logic behind algebra. Simple breakdowns like these empower students to follow the process and replicate it independently.
Verification of Factorization
After factorizing, it's crucial to verify your work. This ensures that the expression you achieved is indeed the correct factorization of the original.
  • Why Verify? - Verification confirms that no errors were made during factorization. It reinforces confidence in using correct methods.
  • The Verification Step - To verify \(5(a - 1)\), distribute the 5 back into the expression: \(5 \times a = 5a\) and \(5 \times (-1) = -5\). Thus, \(5(a - 1)\) results in the original expression \(5a - 5\).
Verification is like double-checking your work in math. It makes sure that all steps logically lead back to the original problem, guaranteeing the soundness of your factorization. Always take a moment to verify for accuracy and understanding.

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Most popular questions from this chapter

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