Book cover for Elementary Algebra

Elementary Algebra

Lynn Marecek

ISBN #9780998625713

1st Edition

6,645 Questions

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32,192 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter introduces the process of factoring algebraic expressions by first extracting the greatest common factor (GCF) using the reverse distributive property and then moving on to more complex methods like factoring by grouping. Additionally, when dealing with quadratic trinomials with a leading coefficient of 1, the method of identifying two numbers that multiply to the constant and add to the linear coefficient proves essential. These techniques not only simplify expressions but also provide fundamental tools for solving equations and modeling real-world scenarios.

Learning Objectives

1

Understand how to find the greatest common factor (GCF) of numbers and algebraic expressions.

2

Demonstrate how to factor a polynomial by extracting the GCF using the reverse Distributive Property.

3

Apply factoring by grouping to polynomials that do not have a common factor across all terms.

4

Factor quadratic trinomials with a leading coefficient of 1 by identifying pairs of numbers that multiply to the constant term and add to the middle coefficient.

5

Utilize factoring techniques to model real-world scenarios such as physical structures and projectile motion.

Key Concepts

CONCEPT

DEFINITION

Greatest Common Factor (GCF)

The largest expression (numeric and variable components) that divides each term of a polynomial without a remainder.

Reverse Distributive Property

The process of rewriting an expression as a product of factors; essentially factoring out a common factor from terms.

Factoring by Grouping

A method used to factor polynomials that do not have a common factor for all terms by grouping terms into pairs and factoring each pair.

Quadratic Trinomial

A three-term polynomial, typically in the form x² + bx + c (with leading coefficient 1), that can be factored into two binomials.

Leading Coefficient

The coefficient of the term with the highest power in a polynomial. In quadratic trinomials of the form x² + bx + c, it is 1.

Example Problems

Example 1

Find the greatest common factor. $$ 8,18 $$

Example 2

Find the greatest common factor. $$ \text { 24, } 40 $$

Example 3

Find the greatest common factor. $$ 72,162 $$

Example 4

Find the greatest common factor. $$ 150,275 $$

Example 5

Find the greatest common factor. $$ 10 a, 50 $$
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Step-by-Step Explanations

QUESTION

Find the GCF of 27x³ and 18x⁴.

STEP-BY-STEP ANSWER:

Step 1: Factor the numerical coefficients into primes. 27 = 3 × 3 × 3 and 18 = 2 × 3 × 3.
Step 2: Write the variable parts in exponent form: x³ and x⁴.
Step 3: Identify the common prime factors and the smallest power of x present in both terms. The common numerical factors are 3 and 3 (i.e., 9) and the variable part is x³.
Step 4: Multiply these together: GCF = 9x³.
Final Answer: 9x³ is the greatest common factor.

Finding the Greatest Common Factor (GCF)

QUESTION

Factor the quadratic trinomial x² + 5x + 6.

STEP-BY-STEP ANSWER:

Step 1: Identify two numbers that multiply to 6 (the constant term) and add to 5 (the coefficient of x).
Step 2: List factor pairs of 6: (1, 6) and (2, 3). Determine which pair sums to 5; here, 2 + 3 = 5.
Step 3: Write the trinomial as the product of two binomials using the found numbers: (x + 2)(x + 3).
Step 4: Optionally, check by expanding the binomials to ensure the product yields the original trinomial.
Final Answer: (x + 2)(x + 3) is the factored form.

Factoring a Quadratic Trinomial (Leading Coefficient 1)

QUESTION

Factor the expression x² + 3x − 2x − 6 by grouping.

STEP-BY-STEP ANSWER:

Step 1: Group terms into two pairs: (x² + 3x) and (−2x − 6).
Step 2: Factor out the common factor from each group: x from the first (x(x + 3)) and −2 from the second (−2(x + 3)).
Step 3: Notice the common binomial (x + 3) in both groups; factor it out.
Step 4: Write the factored form as (x + 3)(x − 2).
Final Answer: (x + 3)(x − 2) is the factored form.

Factoring by Grouping

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Common Mistakes

  • Failing to factor out the entire greatest common factor, including all variable components.
  • Omitting the negative sign when the leading coefficient is negative.
  • Choosing incorrect factor pairs for quadratic trinomials by confusing the requirements of multiplication and addition.
  • Misgrouping terms when attempting to factor by grouping, leading to incomplete factorization.