Book cover for Beginning Algebra

Beginning Algebra

Margaret L. Lial, John Hornsby

ISBN #9780321673480

11th Edition

5,012 Questions

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99,418 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section explores the fundamental concept of factoring, emphasizing that it is the process of expressing a quantity as a product. The focus is on finding the greatest common factor (GCF) for both numbers and variable expressions by using prime factorization and divisibility rules. Additionally, the section demonstrates how to factor polynomials by extracting the GCF via the distributive property and touches on factoring by grouping for more complex expressions. Mastery of these techniques simplifies algebraic manipulations and lays the groundwork for solving equations.

Learning Objectives

1

Understand factoring as the process of writing a quantity as a product, which is the reverse of multiplication.

2

Identify and compute the greatest common factor (GCF) for lists of numbers and variable terms using prime factorization.

3

Apply divisibility rules to efficiently determine the factors of numbers.

4

Factor polynomials by extracting the GCF using the distributive property.

5

Utilize factoring by grouping to simplify more complex algebraic expressions.

Key Concepts

CONCEPT

DEFINITION

Factoring

The process of writing a quantity as a product; essentially, the reverse of multiplication.

Greatest Common Factor (GCF)

The largest factor common to all terms in a set, found by identifying common primes with their smallest exponents.

Prime Factorization

Expressing a number as a product of its prime numbers.

Divisibility Rules

Guidelines that help determine whether one number is a divisor of another (e.g., a number is divisible by 2 if it ends in 0, 2, 4, 6, or 8).

Distributive Property

A property used to factor expressions, allowing a common factor to be factored out of a sum (a*(b + c) = a*b + a*c).

Example Problems

Example 1

Find the greatest common factor for each list of numbers. 40.20 .4

Example 2

Find the greatest common factor for each list of numbers. $50,30,5$

Example 3

Find the greatest common factor for each list of numbers. $18,24,36,48$

Example 4

Find the greatest common factor for each list of numbers. $15,30,45,75$

Example 5

Find the greatest common factor for each list of numbers. $6,8,9$
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Step-by-Step Explanations

QUESTION

Find the GCF for the numbers 30 and 45.

STEP-BY-STEP ANSWER:

Step 1: Express each number in its prime factors. 30 = 2 × 3 × 5 and 45 = 3 × 3 × 5.
Step 2: Identify the common prime factors. Both numbers have 3 and 5 in common.
Step 3: Use the smallest exponent for each common prime (3 appears once in 30, and 5 appears once in both).
Step 4: Multiply the common primes. GCF = 3 × 5.
Final Answer: 15.

Finding the GCF for Numeric Terms

QUESTION

Determine the GCF for the variable terms m^3n^5, m^4n^4, and m^5n^2.

STEP-BY-STEP ANSWER:

Step 1: Write each term showing the exponents for each variable: m^3n^5, m^4n^4, m^5n^2.
Step 2: Identify the common base variables. Both m and n are present in all terms.
Step 3: Choose the smallest exponent for each variable. For m, the smallest exponent is 3, and for n, it is 2.
Step 4: Multiply these common factors. GCF = m^3 × n^2.
Final Answer: m^3n^2.

Finding the GCF for Variable Terms

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Common Mistakes

  • Assuming that factoring each term individually is sufficient, without combining them by factoring out the common factor.
  • Forgetting to use the smallest exponent when determining the GCF for variables.
  • Misapplying or overlooking divisibility rules, leading to incorrect prime factorizations.
  • Believing that a polynomial is fully factored just because each term appears factored, rather than ensuring a common factor is completely extracted.