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Chapter 6: Problem 12

Factor the given expressions completely. $$5 a^{2}-20 a x$$

Short Answer

Expert verified
The fully factored expression is \( 5a(a - 4x) \).

Step by step solution

01

Identify Common Factors

First, look for any common factors that can be factored out from all the terms in the expression. Here, both terms have a common factor of 5 and the variable \( a \).
02

Factor Out the Greatest Common Factor (GCF)

Factor out the greatest common factor identified in Step 1. The expression can be rewritten as \( 5a(a - 4x) \). This is done by dividing each term in the original expression by \( 5a \).
03

Rewrite and Simplify

Now, verify and rewrite the expression to ensure it's fully factored. The completely factored expression is \( 5a(a - 4x) \). This confirms \( 5a \) is indeed the GCF, leaving us with the factor \((a - 4x)\).

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Key Concepts

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

Greatest Common Factor
When dealing with polynomials, the Greatest Common Factor (GCF) is a crucial concept to grasp. The GCF is the largest factor that can divide each term of the polynomial without leaving a remainder. Identifying it is the first step in the factoring process.
  • Begin by examining all terms of the given polynomial. In our example, the expression is composed of the terms \(5a^{2}\) and \(-20ax\).
  • Look for numbers or variables that appear in each term. Here, both terms have the number 5 and the variable \(a\) in common.
  • Therefore, the GCF here is \(5a\). When you factor this out, you're simplifying the expression for easier handling in further algebraic operations.
Polynomial Expressions
A polynomial expression is a mathematical phrase involving sums of powers of variables with coefficients. Understanding the structure helps in manipulating them effectively, especially when factoring. Let's break it down:
  • Polynomials consist of terms, like \(5a^{2}\) or \(-20ax\), where each term is formed by multiplying a coefficient (a number) with variables raised to powers.
  • Identifying possible like terms within an expression can make simplifying much easier. In this case, similar variables \(a\) can be grouped.
  • Factoring polynomials involves expressing them as a product of simpler polynomials or numbers, which reduces their complexity and allows for easier calculations.
Algebraic Manipulation
Algebraic manipulation involves performing operations on algebraic expressions to simplify or solve them. It requires understanding and applying various mathematical concepts, especially when factoring is involved. Here's how it relates to our problem:
  • Start by identifying and extracting the GCF from all terms. This step helps break down the polynomial into more manageable parts.
  • After factoring out the GCF in the example \(5a(a - 4x)\), consider each component. The factors should multiply back to the original polynomial when expanded.
  • The result \((a - 4x)\) is the product of simplification, ensuring that each term in the factor is correctly simplified and verified with the original expression.
The process helps not only in breaking down complex polynomial expressions but also prepares them for use in equations and further algebraic applications.

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